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IN  MEMORIAM 
FLOR1AN  CAJOR1 


PLANE    GEOMETRY 


DEVELOPED  BY  THE 


SYLLABUS  METHOD 


BY 


EUGENE   RANDOLPH    SMITH,    A.M. 

/! 

HEAD  OF  THE  DEPARTMENT  OF  MATHEMATICS,  POLYTECHNIC    PREPARATORY 

SCHOOL,  BROOKLYN,  NEW  YORK 
(FORMERLY  HEAD  OF  THE  DEPARTMENT  OF  MATHEMATICS, 

MONTCLAIR   HIGH    SCHOOL) 


<x>>*i< 


NEW  YORK  •:•  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


COPYRIGHT,  1909,  BY 
EUGENE   RANDOLPH  SMITH. 

ENTERED  AT  STATIONERS'  HALL,  LONDON. 


SMITH   SYL.    GEOM. 

W.  P.     I 


PREFACE 

THE  belief  that  the  proofs  of  Geometry  should  be,  as 
far  as  possible,  worked  out  by  the  pupils,  either  in  class 
discussions  or  individually,  is  becoming  more  widespread 
every  year.  The  day  of  memorizing  proofs  will  soon  be 
past,  and  the  most  efficient  method  for  mental  training 
along  logical  lines  will  be  the  one  generally  adopted. 
This  syllabus  is  written  with  the  hope  of  encouraging 
teachers  to  undertake  Geometry  by  the  "  no  text  "  method. 
The  author  believes  very  decidedly  that  this  method  gives 
a  maximum  of  mental  training  with1  a  minimum  waste 
of  energy. 

The  list  of  theorems  is  based  on  the  latest  reports  of 
the  Mathematical  Associations,  and,  while  much  shorter 
than  that  in  many  of  the  text-books,  it  will  be  found 
sufficient  to  prepare  the  pupils  for  any  of  the  colleges. 
It  contains  all  the  theorems  of  the  "  New  England  List " 
with  a  few  additions  that  simplify  proofs. 

The  order  is  the  development  of  ten  years'  class  use, 
and  will  be  found  different  from  that  of  any  text.  When- 
ever a  theorem  has  seemed  to  be  simplified,  either  in 
content  or  in  proof,  by  making  a  change  in  its  place  in 
the  order  of  theorems,  that  change  has  been  tried  in  class, 
and  has  been  made  permanent  if  it  proved  of  advantage. 
Any  teacher  using  this  book  should  feel  equally  free  to 
make  changes  in  the  order  if  he  is  convinced  that  there 
is  a  decided  advantage  in  the  change. 

3 


4  PREFACE 

The  chapter  on  Logic  has  been  found  of  great  assist- 
ance in  helping  the  pupils  to  think  accurately,  and  it  is 
certain  to  save  more  time  for  a  class  than  its  discussion 
requires. 

The  definitions  and  axioms,  are  given  in  quite  complete 
form,  not  for  assignment  to  the  class,  as  this  part  of  the 
work  should  be  developed  before  the  text  is  given  to  a 
class  for  study;  but  as  a  guide  for  the  teacher,  both  in 
order  and  in  subject-matter,  and  as  a  reference  book  for 
the  pupil.  Good  results  can  be  secured  by  withholding 
the  book  from  the  pupils  until  part,  at  least,  of  the  pre- 
liminary matter  has  been  discussed. 

The  subjects  of  "  existence  "  and  "  betweenness  "  have 
not  been  considered  to  any  great  extent,  as  they  do  not 
seem  worth  the  time  and  effort  required,  except  to  a 
student  of  the  more  advanced  pure  mathematics.  "  Loca- 
tion "  and  "  intersection,"  on  the  other  hand,  are  of  such 
vital  importance  in  considering  the  correctness  of  proofs 
that  they  have  received  some  attention.  The  aim  through- 
out has  been  to  arrange  a  system  of  Geometry  that  should 
be  natural,  reasonably  complete,  and  suitable  to  afford  as 
much  mental  training  as  the  maturity  of  the  pupils  would 
allow.  The  author  has  not  hesitated  to  assume  any  axiom 
that  would  help  more  than  its  presence  would  complicate  ; 
on  the  other  hand,  he  has  left  out  things  that  seemed  to 
require  more  than  they  gave. 

Geometry  itself  has  no  concern  with  measurement  by 
means  of  a  unit.  The  applications  of  Geometry  to  such 
measurement  are,  however,  very  frequent  and  very  im- 
portant, and  while  this  book  presupposes  geometrical 
proofs  to  as  great  a  degree  as  seems  possible  without 
unnecessarily  complicating  the  subject,  there  has  been 
no  attempt  to  draw  a  hard  and  fast  line  of  demarcation 


PREFACE  5 

between  Geometry  and  its  applications.  If  a  teacher 
believes  in  distinguishing  sharply  between  the  different 
branches  of  mathematics,  the  study  of  the  lengths  of  line 
sects  and  the  calculation  of  areas  can  be  put  under  the 
head  of  MENSURATION. 

The  exercises  are  in  two  divisions,  those  under  the 
theorems  and  those  in  general  lists.  The  exercises  under 
the  theorems  have  been  chosen  to  illustrate  the  uses  of 
the  various  theorems,  and  they  should  therefore  be  of 
great  help  to  the  teacher.  The  general  lists  give  the 
pupils  practice  in  finding  for  themselves  what  princi- 
ples underlie  the  proofs.  Probably  no  class  could  finish 
all  the  exercises  in  the  book  in  one  year,  but  the  teacher 
can  easily  choose  those  best  suited  to  his  purposes.  There 
are  several  pages  of  college  examination  questions.  Some 
of  these  are  duplicates  of  exercises  scattered  through  the 
book,  but  the  differences  in  wording,  as  well  as  the  desire 
to  let  students  know  what  type  of  questions  examiners 
ask,  has  prompted  leaving  them  in  the  book. 

This  book  has  been  written  with  little  reference  to  the 
order  and  methods  of  other  texts,  for  it  is  a  compilation 
that  has  grown  naturally  from  class  work.  •  The  author 
is,  of  course,  indebted  for  many  of  the  ideas  used  to 
numerous  works  on  mathematics  and  its  pedagogy,  but 
in  many  cases  it  is  now  impossible  to  tell  from  what 
source  the  suggestion  first  arose.  He  wishes,  however, 
to  acknowledge  his  special  indebtedness  to  Dr.  William  H. 
Metzler  of  Syracuse  University,  for  assistance  and  encour- 
agement in  the  writing  of  this  book. 

EUGENE   R.   SMITH. 


SUGGESTIONS   TO   TEACHERS 

Preliminary  Definitions  and  Axioms.  Do  not  assign 
these  to  the  class  to  read  until  after  they  have  been  thor- 
oughly discussed  in  class.  As  far  as  possible,  let  the 
pupils  frame  the  definitions  for  themselves,  and  lead  up 
naturally  to  the  simple  deductions  from  them  (starred  in 
the  text),  so  that  the  pupils  can  begin  to  discover  these 
truths  from  the  very  beginning. 

Every  word  of  this  part  need  not  be  digested  before 
going  on  to  the  theorems ;  but  before  any  new  work  is 
undertaken  the  teacher  should  make  sure  that  the  pupils 
have  a  perfect  grasp  of  the  particular  facts  to  be  used  in 
developing  the  new  matter.  Certain  parts  can  be  touched 
upon  lightly  on  the  first  reading,  and  cleared  up  thor- 
oughly just  before  being  used.  Make  haste  slowly. 

Theorems.  Develop  the  very  difficult  ones  in  class  by 
the  question  and  answer  method  of  analysis ;  assign  those 
less  difficult  for  outside  preparation  or  to  be  worked  out 
in  class,  either  at  the  board  or  at  the  pupils'  seats.  Clear 
up  the  work  frequently  by  review  recitations  covering 
the  theorems  recently  done.  Use  a  great  deal  of  oral 
work,  asking  for  every  viewpoint  on  both  new  and  old 
work.  Never  tell  the  pupils  what  to  do;  ask  questions  so 
framed  that  the  pupils  are  made  to  think.  Summarize 
frequently,  and  by  the  use  of  exercises  cultivate  the  pupils' 
power  to  choose  the  correct  method. 

6 


SUGGESTIONS   TO   TEACHERS  7 

Notebooks.  The  author  advises  that  the  pupils  be  re- 
quired to  keep  notebooks  in  which  they  write  out  all 
the  theorems,  unless  the  conditions  under  which  the 
teacher  is  working  make  it  impossible.  All  necessary 
corrections  can  be  noted  by  the  teacher  and  made  by  the 
pupil,  after  which  the  theorems  can  be  filed,  and  so  be- 
come at  the  end  of  the  course  a  complete  reference  book 
compiled  by  the  pupil.  A  loose-leaf  system  will  be  found 
convenient. 

Time  Required.  Do  not  hurry  the  early  work.  What- 
ever time  is  used  in  obtaining  a  thorough  understanding 
of  the  foundations  and  of  the  methods  will  be  more  than 
repaid  by  increased  speed  later  on. 

The  first  quarter  of  the  year's  course  should  be  spent 
on  the  work  up  to  the  section  on  Locus  in  Book  I,  and 
if  this  assignment  is  not  quite  finished,  no  great  harm  will 
be  done. 

The  second  quarter  should  almost,  if  not  quite,  finish 
the  second  book. 

The  third  quarter  should  take  about  to  the  construc- 
tions of  the  fourth  book. 

The  last  quarter  should  finish  the  syllabus  and  leave 
at  least  a  month  for  general  review  and  additional  exer- 
cise work. 

This  is  an  estimate  of  a  fair  average  rather  than  of  an 
excellent  record,  and  many  classes  can  do  much  better ; 
it  supposes  that  a  good  deal  of  original  work  on  exer- 
cises, and  almost  daily  oral  work  on  uses  of  theorems, 
methods  of  attack,  and  other  important  topics  have  been 
taken. 

Logic.  Make  frequent  use  of  the  section  on  Logic.  Ask 
the  pupils  to  state  the  converse  and  the  contraposite  of 
very  many  of  the  theorems,  and  to  discover  whether  these 


8  SUGGESTIONS   TO   TEACHERS 

are  new  theorems.  In  this  way  the  proofs  of  many  theo- 
rems and  of  many  excellent  exercises  will  be  found. 

Limits  and  the  Incommensurable  Case.  In  common 
with  many  of  the  best  educators,  the  author  believes  that 
the  proofs  of  these  theorems  are  too  difficult  for  imma- 
ture minds.  He  therefore  recommends  that  the  Appendix 
proofs  be  omitted,  but  that  some  explanation  of  the  possi- 
bility of  the  incommensurable  case  be  given. 

Methods  of  Attack.  It  is  the  experience  of  the  author 
that  the  methods  of  pure  logic  and  the  classification  and 
elimination  method  are  the  ones  of  principal  importance 
for  proving  theorems.  The  so-called  "  Indirect  Method  " 
and  "  Reductio  ad  Absurdum "  are  unnecessary  if  logic 
is  understood.  "  Intersection  of  Loci,"  while  following 
directly  from  the  classification  method, 'has  its  uses,  and 
the  analysis  method  for  constructions  is  invaluable.  The 
analyses  of  the  sample  theorems  will  serve  as  a  guide  to 
the  class  method  of  discovering  the  proof  for  a  theorem. 


CONTENTS 
BOOK   L—  RECTILINEAR   FIGURES 

PAGES 

I.     LOGIC 15-21 

II.     POINTS,  LINES,  AND  SURFACES         ....  22-27 

III.  EQUALITY 28-31 

IV.  ANGLES 32-38 

V.    POLYGONS 39-41 

VI.     INEQUALITIES 42 

VII.     PROPOSITIONS     .        . 43-46 

ORAL  AND  REVIEW  QUESTIONS       .        .        .        .  46, 47 

VIII.     TRIANGLE  THEOREMS        .        .                 .        .        .  48-64 

IX.     PARALLELS  AND  PARALLELOGRAMS         .        .        .  65-74 

X.     Loci  AND  CONCURRENCE 75-79 

SUMMARY  OF  PROPOSITIONS     .        .        .                 .  79, 80 

ORAL  AND  REVIEW  QUESTIONS       .        .        .        .  80, 81 

GENERAL  EXERCISES 81-86 

BOOK   II.  — CIRCLES 

I.     CIRCLE  DEFINITIONS 87-93 

ORAL  AND  REVIEW  QUESTIONS       ....  93 

II.     CONSTRUCTIONS 94-100 

III.  CIRCLE  THEOREMS 101-109 

IV.  CONSTRUCTIONS  (Usi\G  CIRCLE  THEOREMS)  .        .  110-111 

SUMMARY  OF  PROPOSITIONS 112, 113 

ORAL  AND  REVIEW  QUESTIONS        ....  113 

GENERAL  EXERCISES 114-118 


BOOK   III.  — EQUIVALENCE   AND   AREA 

I.     DEFINITIONS  AND  FORMULAS 119-122 

II.     THEOREMS 123-132 

III.    CONSTRUCTIONS 133-134 

SUMMARY  OF  PROPOSITIONS 134-136 

ORAL  AND  REVIEW  QUESTIONS       ....  136, 137 

GENERAL  EXERCISES 137, 138 

9 


10  CONTENTS 

BOOK   IV.  — SIMILAR   FIGURES:    PROPORTIONS 


I.     RATIO  AND  PROPORTION 
SUMMARY 

• 

.     139-141 
.     141,  142 

II.     PROPORTIONAL  SECTS  . 

. 

.     145,144 

III.     SIMILAR  FIGURES 

. 

.      ...        .     145-151 

IV.     CONSTRUCTIONS     . 

. 

.     152-154 

SUMMARY  OF  PROPOSITIONS, 

.     154,155 

ORAL  AND  REVIEW  QUESTIONS  . 

.     155,  156 

GENERAL  EXERCISES    . 

. 

156-158 

BOOK   V.  — REGULAR   POLYGONS   AND   CIRCLES 

I.     THEOREMS     .        . 159-163 

II.     CONSTRUCTIONS 164, 165 

SUMMARY  QF  PROPOSITIONS        .        .        .        .        .  165 

ORAL  AND  REVIEW  QUESTIONS  .....  166 

GENERAL  EXERCISES 166, 167 

GENERAL 

THE  FORMULAS  OF  GEOMETRY 168-170 

COLLEGE  EXAMINATION  QUESTIONS 171-182 

APPENDIX 

CONTRAPOSITE  LAW 183 

LAW  OF  CONVERSE 183 

PROOFS    OF    THE    OBVERSE    AND    CONVERSE    OF    A    SINGLE 

STATEMENT 184 

GENERAL  CONDITION    .        .        .        .     .   .        .        .                 .  185 

AXIOMS ,.  185 

SYMMETRY 185 

POSITIVE  AND  NEGATIVE  SECTS  AND  ANGLES  ....  186 

DISTANCE 186 

LIMITS 187 

INCOMMENSURABLE  CASE 189 

SIMILAR  FIGURES 190 

THE  EVALUATION  OF  Pi                                                      .        .  190 


INDEX 


TOPIC   REFERENCES   AND   ABBREVIATIONS 


TERM          ABBREV.    ART. 
Acute  angle (d)  63,  80 

Addition  of  polygons  226,  228 

Adjacent  (angles)  Adj.  46 
Alternate  (angles)  Alter.  115 
Altitude  Alt.  81,  245 

Ambiguous  Case  130 

Analysis  Anal.  92,  94-5, 178 

Angle  (Sect.  IV)        Z          44,  45 

of  polygon  47 

Antecedent  258 

Apothern  Apv          313 

Approach  (as  a  limit)  =  318 
Arc  161 

Area  69, 160,  236,  245 

Arm  44 

Axiom  Ax.  14, 23,  27,  38-42, 

68,82,119,321,313 
Axis  344 


76,  79, 135 
42 


Bisect 
Broken  line 


Center  160,  313, 344 

Center  line  or  center  sect        168 

Central  angle  188 

Centroid  157 

Chord  161 

Circle  O           160 

Circumcenter  152 

Circumference  Circum.    160 

Circumscribed  Circumsc.  165 

Classification  88 


TERM  ABBREV.    ART. 

Closed  line  27 

Collinear  164 

Commensurable  Commen.  203 
Common  chord  172 

Compass  176 

Complements  Comp.   61,  193 

Composition  261 

Concave  polygon  73 

Concentric  circles  198 

Conclusion  Concl.  2,  85 

Concurrent  151 

Concyclic  '  218 

Condition  Cond.    2,  85 

Conditional  statements  2 

Congruent  ^          32-3 

Consequents  258 

Constant  318 

Construction  Const.     84 

Contact  173 

Continued  proportion  259 

Contraposite  Cont.    6,  8,  339 

Converse  Conv.  4, 9,  340-1 

Convex  polygon  73 

Corollary  Cor.         84 

Corresponding  angles  Cor.  A  115 
Corresponding  parts  Cor.  Pts.  93 
Cross  polygon  73 

Curved  line  28 


Decagon 

Definition 

Degree 


75 

Def.        13 
Deg.,°  134,206 


11 


INDEX 


TERM           ABBREV.    ART. 

TERM           ABBREV. 

ART. 

Describe                                     179 

Intersection                      68, 

167,  170 

Determine  (line)              89,  98,  110 

Inversely  proportional  Inv. 

267 

Diagonal                                     72 

Isosceles                      Isos. 

79 

Diameter                                   161 
Difference                                   47 
Dimensions                          16,  17,  18 

Law  of  contraposite 
Law  of  converse 

8 
10 

Distance                       Dist.      346 

Legs                                      79,  80,  135 

Division  (prop.),  (sect)  261,  279,  306 

Length 
Limit 

236 
318 

Elimination                                90 

Line 

18 

Equality  (Sect.  Ill)    =         32-38 

Linear 

236 

Equiangular             Equiang.     74 

Location 

96,  110 

Equilateral               Equilat.     74 

Locus 

146 

Equimultiples                           262 

Logic  (Sect.  I)             1-11, 

339-342 

Equivalence                  =        34,  225 
Escribed                                     166 
Evaluation  of  Pi                       350 
Excenter                                     155 

Major  arc 
Mean  and  extreme  ratio 
Mean  proportion 

Means 

193 
306 
259 
258 

Explements                  Exp.    56,  193 

Measurement                 203, 

206,  236 

Exterior  angles            Ext.  A  71 

Median 

81 

Ext.  Interior  angles  Ext.  Int.  115 

Mensuration 

Preface 

Extremes                                   258 

Methods 

90,  147 

Figure                           Fig.         29 

Minor  arc 

193 

Formulas                 23Q,  245,  246,  338 

Minute                           ' 

134,  206 

Fourth  proportional                 258 

Negative  converse 

6 

General  and  special  cases         11 

Negative  sects  and  angles 

345 

Geometry                                    12 

Negative  statement 

5 

TTrtl-C                                                                                      >i  n 

Numerical 

242 

Halt                                             42 

Harmonic  division                   279 

Oblique 

58 

Hexagon                                     75 

Obtuse  angle  (d) 

63,80 

Homologous                               93 

Obverse                       Obv. 

5,341 

Hypotenuse                  Hyp.      80 

Octagon 

75 

Hypothesis                               2,  85 

Order  of  proof 

87 

Incenter                                     154 

Orthocenter 

156 

Incommensurable    Incom.  203,  348 

Parallel                        II 

118 

Inequality  (Sect.  VI)  Ineq.       83 

Parallelogram              O 

135 

Inscribed                      Insc.     165 

Pencil  of  lines 

275 

Inscribed  angle                        188 

Pentagon 

75 

Interior  angles             Int.  A 

Perigon 

54 

of  polygon,  71  ;  Us,           115 

Perimeter                  Per. 

69 

INDEX 


13 


TERM           ABBREV.    ART. 

TERM           ABBREV.     ART. 

Perpendicular             JL             58 

Similar                  ~        282,  289,  349 

Pi                                  TT            324 

Solid                                            16 

Plane                                          30 

Space                                          15 

Plane  geometry                          31 

Special  cases                             11 

Point                            Pt.           19 

Square                          Q        135,  229 

Point  cutting  sect                     279 

Stand  on                                    188 

Point  of  tangency                 167,  173 

Straight  angle              St.  Z       49 

Polygon  (Sect.  V)                     69 

Straight  edge                             176 

Positive  sects  and  angles         345 

Straight  line                               22 

Postulate                     Post.      176 

Subtend                                    188 

Problem                       Prob.       84 

Subtraction                              227-8 

Projection                   Proj.      243 

Sum                              +      47,  227-8 

Proof                                    86-7,  260 

Summary   89,    158,  223,  256,  274, 

Proportion  (al)             Prop.  202,  258 

310,  336 

Proposition  (Sect.  VII)            84 

Superimpose                              41 

Pythagorean  proposition        240 

Supplements                Sup.     51,  193 

Quadrant  of  arc                        192 
Quadrilateral               Quad.     75 

Surface                                       17 
Symmetry                                 344 

Radius                      Rad.;  r  161,313 

Tangent                       Tang.  167,173 

Ratio                                         202 

Terms                                          202 

Rectangle                     CH       135,229 

Theorem                      Th.          84 

Rectilinear                                 29 

Third                                          42 

Reflex                                         63 

Third  proportional                   259 

Regular                        Reg.        74 

Touch                                        173 

Representation                          21 

Transversal                              '115 

Rhomboid                                 135 

Transverse  set       Trans  set  115 

Rhombus                                    135 

Trapezium                                  135 

Right  angle  (d)            Rt.  Z     58,80 

Trapezoid                    Trap.      135 

Scalene                                       97 

Triangle                       A      75,  77-81 

Trisect                                        42 

Secant                                        167 

Second                          "         134,  206 

Unit                                          236 

Sect                                             25 

Sector                                        187 

Variable                      Var.       318 

Segment                                   25,  187 

Vertex  angle               Vert.  Z  78 

Semicircle                 Semi  O     193 

Vertical  angle  of  triangle         78 

Semicircumference  Semicir.  193 

Vertical  angles            Vert.  A  48 

Side                         S.               70 

Vertices  of  polygons                 70 

SUMMARY   OF   GEOMETRICAL   SIGNS 


> 


plus,  sign  of  addition 
minus,  sign  of  subtraction 
times,  sign  of  multiplication 
,  /,  :  ,  divided'by,  sign  of  division 
square  root  sign 
is  (or  are)  equal,  or  equivalent 

is  not  equal,  or  equivalent,  to 
is  identical  to 
is  congruent  to 
approaches  as  a  limit 
is  similar  to 
is  greater  than 

is  not    reater  than 


<  is  less  than 

«jC  is  not  less  than 

II  is  parallel  to 

_L  is  perpendicular  to 

Z  or  2jL  angle 

A  triangle 

O  parallelogram 

n  rectangle 

Q  square 

O  circle 

^  arc 

.•.  therefore 

• .  •  because,  since 


The  signs  for  figures  become  plural  by  the  addition  of  s,  often  within 
the  sign,  as  U)  for  rectangles. 


OTHER   ABBREVIATIONS 

eq.  -f  eq.,  for  "equals  plus  equals,"  and  similarly  for  the  other  axioms. 

2  s.  incl.  Z,  for  "two  sides  and  the  included  angle,"  etc. 

In  referring  to  propositions  or  axioms  as  authority,  it  is  permissible  to 
use  any  easily  understood  abbreviation  for  the  theorem.  The  ones  given 
above  are  good  examples  of  abbreviations  that,  if  used  correctly  in  a 
theorem,  could  hardly  be  misunderstood. 


PLANE  GEOMETRY 

BOOK   I.     RECTILINEAR   FIGURES 

SECTION  I.    LOGIC 

1.  Need  of  Logic.     In  all  discussions,  more  or  less  logic 
is  used,  though  often  unknowingly.     For  example,  if  a 
person,  noticing  that  a  flag  is  hanging  limp  on  a  pole,  says, 
"  There  is  no  wind  to-day,"  that  person  has  reasoned  as 
follows : 

"  If  the  wind  were  blowing,  the  flag  would  be  waving. 
But  the  flag  is  not  waving,  so  the  wind  is  not  blowing." 

Because  such  reasoning  is  so  instinctive  that  we  seldom 
realize  the  steps  through  which  the  mind  arrives  at  the 
conclusion,  the  need  of  a  knowledge  of  the  most  important 
laws  of  logic  is  not  fully  appreciated. 

In  Geometry,  the  same  kind  of  reasoning  must  be 
applied,  and  as  instinct  with  regard  to  the  figures  to  be 
studied  is  not  reliable,  a  brief  study  of  the  most  commonly 
applied  laws  of  logic  is  necessary. 

2.  Conditional  Statements.  Practically  all  statements  are 
conditional,  though   the  condition  is  sometimes  implied 
rather  than  stated.     Even  the  statement,  "It  rains,"  de- 
pends upon  the  place  and  time. 

When  a  statement  asserts  that  the  truth  of  one  thing- 
assures  the  truth  of  another  thing,  the  one  upon  which  the 

15 


16  LOGIC 

other  depends  is  the  condition  (or  hypothesis),  the  other 
the  conclusion  of  the  statement. 

EXAMPLES: 

CONDITION  CONCLUSION 

If  it  rains  to-day,  I  will  stay  at  home. 

If  a  triangle  has  two  equal  sides,  the  opposite  angles  are 

equal. 

A  horse  has  four  legs. 

If  John  goes  down  town,  I  will  go  with  him. 

NOTE.  In  the  second  example,  the  word  "triangle"  simply  in- 
dicates the  figure  in  which  the  reasoning  is  to  be,  and  for  the  purposes 
of  Geometry  it  does  not  need  to  be  considered  as  part  of  either  the 
condition  or  the  conclusion.  The  terms  used  in  this  illustration  have 
not  been  defined  as  yet,  but  they  are  common  enough  to  be  familiar 
to  most  students. 

3.  Four  Related  Statements.  There  are  four  related 
conditional  statements  which  can  be  formed  from  two  pos- 
sible truths  used  in  the  positive  and  the  negative.  Let  the 
two  possible  truths  be  "  it  rains,"  and  "  the  sidewalk  is 
wet";  then  the  following  related  statements  might  be  made 
(with  no  consideration  at  present  as  to  whether  they  are 
true  statements  or  not)  : 

(1)  If  it  rains,  the  sidewalk  will  be  wet. 

(2)  If  the  sidewalk  is  wet,  it  has  rained. 

(8)  If  it  does  not  rain,  the  sidewalk  will  not  be  wet. 
(4)  If  the  sidewalk  is  not  wet,  it  has  not  rained. 

If  the  two  possible  facts  were  represented  by  A  and  B, 
where  A  and  B  stand  for  any  two  possibilities  such  that 
one  might  depend  upon  the  other,  these  statements  could 
be  written  as  follows  : 

(1)  If  A,  then  B.  (3)   If  not  A,  then  not  B. 

(2)  If  B,  then  A.  (4)  If  not  J5,  then  not  A. 


LOGIC  17 

4.  Converse.  The  first  and  second  statements  have  the 
condition  of  each  the  same  as  the  conclusion  of  the  other  ; 
or,  the  condition  and  conclusion  of  either  interchanged  will 
the  other.  They  are  called  converse  to  each  other. 


5.  Negative  or  Obverse.     The  first  and  third  have  the 
condition  and  conclusion  of  either  the  negative  of  the 
condition  and  the  conclusion  of  the  other  ;  or,  the  condition 
and  conclusion  of  either  made  negative  will'  form  the  other. 
They  are  called  negative  or  obverse  to  each  other. 

6.  Negative  Converse  or  Contraposite.      The  first  and 
fourth  have  the  condition  of  each  the  negative  of  the 
conclusion  of  the  other  ;  or,  the  condition  and  the  conclusion 
of  either  interchanged  and  made  negative  will  form  the  other. 
They  are  called  negative  converse  or  contraposite  to  each 
other.     The  reason  for  the  name  "  negative  converse  "  is 
evident  from  the  definitions  of  these  three  relations  in  §§ 
4,  5,  and  6. 

7.  Exercises. 

1.  Write  three  conditional  statements,  separate  each  into  condition 
and  conclusion,  and  write  the  converse,  negative,  and  negative  con- 
verse of  each. 

2.  What  relation  have  statements  2  and  3  ?  2  and  4?  3  and  4? 

3.  What  relation  to  the  original  statement  has 
(a)  the  converse  of  its  negative? 

(Z>)  the  converse  of  its  negative  converse? 

(c)  the  negative  of  its  negative  converse? 

(d)  the  negative  of  the  converse  of  its  negative  converse? 

8.  Contraposite  (or  Negative  Converse)  Law.     The  con- 
traposite of  any  true  conditional  statement  is  also  true. 

NOTE.      This  law  is  discussed  more  fully  in  the  Appendix,  §  339. 
SMITH'S  SYL.  PL.  GEOM.  —  2 


18  LOGIC 

It  is  true  also  that  the  negative  converse  of  any  false 
statement  is  false,  although  this  is  of  less  importance  than 
the  former  statement. 

The  most  common  way  of  using  this  method  of  reason- 
ing is  to  make  some  true  statement  that  a  certain  con- 
clusion follows  a  certain  condition ;  then,  on  finding  that 
the  conclusion  is  no't  true,  to  say  that  the  condition  is  not 
true.  The  statement  about  the  flag  in  §  1  was  an  example 
of  this.  Anoth'er  example  of  a  very  self-evident  kind  is: 
One  is  looking  for  a  book  with  a  green  cover,  and  finding 
a  book  the  cover  of  which  is  not  green,  knows  at  once 
that  the  book  is  not  the  book  wanted.  Definitely  put, 
the  argument  would  be  : 

The  book  wanted  has  a  green  cover. 

This  book  has  not  a  green  cover,  so  it  is  not  the  right 
book. 

Contraposite  reasoning  is  probably  the  most  common 
kind  of  reasoning,  for  almost  every  one  uses  it  many  times 
daily.  It  is,  however,  used  more  or  less  unconsciously 
in  daily  life,  while  in  Mathematics  and  Science,  and  in 
many  other  places  where  the  reasoning  needs  to  be  per- 
fectly accurate,  one  must  know  definitely  just  what  steps 
are  being  taken,  in  order  that  no  error  can  creep  in. 

EXERCISES.  Suppose  that  if  A  is  true,  B  is  also  true ;  what  other 
statement  regarding  A  and  B  is  known  to  follow?  Write  five  true 
statements,  then  see  if  their  coutraposites  are  also  true.  See  if  their 
converses  are  true. 

9.  Truth  of  the  Negative  and  the  Converse  of  a  True 
Statement.  The  negative  and  the  converse  of  a  single  true 
statement  are  not  necessarily  true.  They  are  either  both 
true,  or  both  false,  for  they  are  negative  converse  to  each 
other.  The  discussion  of  when  they  are  true  will  be 


LOGIC  19 

taken  up  in  §  10.     The  following  examples  will  show  that 
they  are  not  always  true : 

uAn  apple  tree  has  leaves."     It  does  not  follow  that 

(1)  If  it  is  not  an  apple  tree,  it  has  not  leaves. 

(2)  If  it  has  leaves,  it  is  an  apple  tree. 

"This  desk  is  made  of  wood."     It  does  not  follow  that 

(1)  If  it  is  not  this  desk,  it-  is  not  made  of  wood. 

(2)  If  it  is  made  of  wood,  it  is  this  desk. 

"  A  donkey  has  a  head."  It  does  not  follow  that  be- 
cause you  have  a  head  you  are  a  donkey. 

WARNING.     Never  assume  that   the  negative  or  the  converse  of  ONE 
true  statement  is  also  true. 

10.  Law  of  Converse.  If  conditional  statements  such 
that  their  conditions  cover  all  possibilities,  and  no  two 
conclusions  can  be  true  at  once,  are  true,  then  the  con- 
verses of  those  statements  are  also  true. 

NOTE.     This  law  is  discussed  more  fully  in  the  Appendix,  §  340. 

This  law  will  be  understood  when  it  is  applied  to  defi- 
nite cases,  but  the  following  examples  make  its  meaning 
somewhat  clearer. 

(1)  If  A  is  true,  X  is  true. 

If  A  is  not  true,  X  is  not  true. 

These  two  conditions  (true  and  untrue)  cover  all  pos- 
sibilities, and  the  two  conclusions  (true  and  untrue)  are 
such  that  they  cannot  both  be  true  at  the  same  time,  so  the 
converses  are  also  true  ;  i.e. 

If  X  is  true,  A  is  true. 

If  X  is  not  true,  A  is  not  true. 

(2)  HA>BI  X>Y. 
If  A  =  B,  x  =  Y. 

If  A<B,X<  Y. 

These  three  conditions  (>,=,<)  cover  all  possibilities, 


20  LOGIC 

and  but  one  of  the  conclusions  (>,=,<)  can  be  true  at 
once,  so  the  converses  are  also  true  ;  i.e. 
If  X>  F,  A  >B. 


If  X  <  F,  A  >  B. 

11.    General  and  Special  Cases. 

NOTE.  This  need  not  be  read  until  the  pupil  is  ready  to  begin 
the  theorems. 

Anything  known  of  mankind  is  known  of  each  man 
separately;  but  anything  known  of  certain  men  only,  would 
not  necessarily  be  true  of  all  men.  If  it  is  true  for  each 
man  in  existence,  it  is  true  for  all  mankind.  In  other 
words,  anything  known  of  a  class  as  a  whole,  or  of  all 
members  of  the  class,  is  known  to  be  a  characteristic  of 
that  class,  both  as  a  whole  and  by  individuals.  On  the 
other  hand,  anything  known  of  part  of  a  class  is  not 
known  of  the  class  as  a  whole,  or  of  other  members  of 
that  class. 

So  in  Geometry,  proofs  should  be  made  for  the  general 
case  whenever  that  is  possible,  and  when  that  does  not 
seem  possible,  the  proof  should  be  worked  for  each  of  the 
different  cases  separately.  For  example,  in  working  with 
triangles,  the  triangle  used  should  always  be  one  about 
which  no  assumption  (other  than  the  given  of  the  theo- 
rem) is  made.  The  triangle  should  neither  be  isosceles, 
nor  be  assumed  to  have  any  certain  sized  angle,  such  as  a 
right  or  acute  angle.  It  is  better  to  draw  the  triangle  in 
the  figure  so  that  it  does  not  even  appear  to  have  any 
special  characteristic,  or  the  one  studying  the  figure  may 
carelessly  assume  that  the  characteristic  which  the  figure 
appears  to  have  really  belongs  to  it. 
'  In  theorems  where  it  does  not  seem  possible  to  find  a 


LOGIC  21 

proof  for  the  general  case  (and  these  are  comparatively 
rare),  it  is  necessary  to  prove  enough  cases  to  cover  all 
possibilities  in  order  that  the  general  case  may  be  known; 
for  example,  a  proof  for  right,  acute,  and  obtuse-angled 
triangles  would  be  true  for  all  triangles.  In  the  same 
way,  a  proof  for  triangles  having  three  equal  sides,  two 
equal  sides,  and  no  equal  sides,  would  be  true  for  all 
triangles. 

Sometimes  a  proof  is  true  only  for  a  special  case  on  ac- 
count of  points  or  line's  that  are  added  to  the  figure  and 
are  assumed  to  lie  in  certain  positions,  when  as  a  matter 
of  fact  they  can  equally  well  lie  in  other  positions.  If 
points  or  lines  are  added  to  a  figure,  the  proof  must  hold 
for  all  possible  positions  in  which  they  can  lie.  This  is 
discussed  in  more  detail  in  §  110. 


SECTION  II.     POINTS,  LINES,  AND  SURFACES 

12.  Geometry.    This  subject  studies  points,  lines,  and 
figures  formed  by  them.     It  proves  facts  about  the  figures, 

*and  uses  as  a  basis  for  the  reasoning  definitions  and  axioms. 

13.  Definitions.    A  definition  is  such  a  description  of  the 
thing  defined  as  will  distinguish  it  from  all  other  things; 
it  might  be  said  to  be  an  agreement  as  to  what  a  term 
shall  be  used  to  indicate.     Some  things  are  of  such  simple 
nature  that  it  is  difficult,  if  not  impossible,  to  define  them 
in  terms  still  simpler,  and  in  such  cases  an  explanation  in 
regard  to  them  may  well  take  the  place  of  a  definition. 

14.  Axioms.    A    truth   that   is    taken    as   one   of    the 
foundation  facts  of  a  subject  is  called  an  axiom.     It   is 
often  defined  as  a  truth  so  simple  that  it  cannot  be  de- 
rived   from    truths    still    simpler ;     but    for   Elementary 
Geometry  this  is  not  strictly  true  (Appendix,  §  343).     It 
will  be  found  that  the  axioms  of  Geometry  are  facts  so 
self-evident  that  there  is  no  doubt  as  to  their  truth. 

15.  Space.    The  space  in  which  everything  exists  is,  as 
far  as   experience    shows,  unlimited.     At   any  rate,   the 
space  studied  in  Elementary  Geometry  (sometimes  called 
Euclidean  Space)  is  unlimited.     Space  is  evidently  divis- 
ible, for  all  bodies  occupy  portions  of  space. 

16.  Solids.    Any  limited  portion  of  space,  such  as  the 
space  occupied  by  any  body,  is  called  —  irrespective    of 
the  nature  of  the  body  which  may  occupy  it  —  a  geometric 

22 


POINTS,   LINES,   AND  SURFACES  23 

solid,  or   simply  a  solid.     Solids   are   said  to  have   three 
dimensions:  length,  breadth,  and  thickness. 

17.  Surfaces.    That  which  separates  one  portion  of  space 
from  an  adjoining   portion   is  called  a  surface.     If  two 
adjoining  lots  are  considered  as  extending  down  into  the 
ground  so  that  any  distance  down  there  is  still  a  boundary 
between  the  lots,  that  boundary  is  a  surface.     It  evidently 
does  not  occupy  space,  for  any  particle  of  soil  belongs  to 
one  lot  or  to  the  other,  yet  there  is  a  distinct  boundary 
such  that  all  on  one  side  of  it  belongs  to  one  lot,  while  all 
on  the  other  side  of  it  belongs  to  the  other  lot. 

NOTE.  The  terms  "side,"  "between,"  "within,"  "outside,"  will 
be  used  in  this  syllabus  in  their  ordinary  meaning  without  any 
attempt  to  define  them  geometrically. 

Another  example  of  a  surface  is  the  outside  of  any  object, 
as  a  box.  It  separates  the  space  occupied  by  the  object 
from  the  space  outside  the  object,  but  itself  occupies  no 
space. 

A  surface  may  be  limited  or  unlimited  in  extent,  and 
may  have  limited  portions  ;  it  is  said  to  be  two  dimen- 
sional, having  length  and  breadth. 

18.  Lines.    That  which  separates  one  portion  of  a  sur- 
face  from   an   adjoining   portion   is   called  a  line.     The 
surface  boundary  between  two  lots  is  a  line.     The  line 
occupies  no  space,  yet  definitely  divides  one  lot  from  the 
other. 

A  line  may  be  indefinite  in  extent,  but  has  limited 
portions ;  it  is  said  to  be  one  dimensional,  having  length 
only. 

19.  Points.    That   which   separates    one   portion   of  a 
line  from  an  adjoining  portion  is  called  a  point.     If  four 
lots  come  together  in  what  is  ordinarily  called  a  corner, 


24  RECTILINEAR  FIGURES 

that  corner  is  a  point.  It  is  a  place,  for  it  is  fixed  in 
position,  but  it  occupies  no  space,  and  no  portion  of  the 
surface.  There  is  evidently  no  unowned  spot  at  the 
corner,  for  all  four  lots  extend  to  the  point,  leaving  no 
unoccupied  surface. 

A  point  has  no  dimensions  and  is  indivisible;  *it  has 
position  only. 

20.  These  definitions  might  have  been  taken  in  reverse 
order;    that    is,    starting   with    the    point.      A    moving 
point  passes  through  (or  describes)  a  line ;  a  moving  line 
usually  describes   a  surface;    a  moving   surface    usually 
describes  a  solid.     It  can  be  seen  that  it  might  be  possible 
for  a  line  to  move  along  itself  in  such  a  way  as  not  to 
describe  a  surface,  and  for  a  surface  to  fail  to  describe  a  solid. 

21.  Representations.     Points,    lines,    and   surfaces    are 
represented  by  various  things.     The  corner  of  a  lot  may 
be  marked  by  a  fence  post,  and  the  line  by  a  fence,  but  it 
is  evident  that  the  post  and  the  fence  are  not  the  point 
and  the  line.     So  in  Geometry,  a  pencil  or  chalk  mark 
may  be  used  to  represent  a  point  or  a  line,  but  it  is  under- 
stood that  they  are  used  only  to  represent  the  things,  and 
that  they  are  not  the  points  and  lines  themselves. 

22.  Straight  Lines.    The  straight  line  is  the  most  im- 
portant and   the   most  familiar  kind   of  line.     It  is  not 
easy  to   define,  but   some   discussion  of   it  is  necessary. 
Different  straight  lines,  or  parts  of  the  same  straight  line, 
need  have  but  two  points  in  common  to  coincide  through- 
out,  irrespective  of   how  the   lines   are   placed  in    other 
respects,  and  this  characteristic  of   straight  lines  is   the 
foundation  of  the  definition. 

The  two  following  are  probably  the  best  definitions  of 
the  straight  line: 


POINTS,  LINES,   AND   SURFACES  25 

(0)  A  line  such  that  any  part  will,  however  placed,  lie 
wholly  on  any  other  part,  if  its  extremities  are  made  to 
fall  upon  that  other  part,  is  called  a  straight  line. 

(5)  A  line  that  is  determined  by  any  two  of  its  points 
is  a  straight  line. 

The  word  "  determined  "  here  means  that  the  line  is 
distinguished  from  any  other  line  of  the  same  kind  by  the 
fact  that  it  goes  through  these  particular  two  points  ;  or,  in 
other  words,  that  it  is  the  only  straight  line  through  the 
two  points. 

23.  However  the  definition  is  worded,  the  fundamental 
fact  about  straight  lines  is  the  following : 

Straight-line  Axiom.  Through  two  points  but  one 
straight  line  can  pass. 

An  illustration  to  show  that  this  distinguishes  the 
straight  line  from  all  others  can  be  made  by  folding  a 
sheet  of  paper  smoothly  and  drawing  any  line  through 
two  points  on  the  edge  formed,  inside  the  fold.  If  the  line 
is  drawn  in  ink,  and  the  sheet  is  folded  firmly  together,  a 
second  line  exactly  like  the  first,  but  in  the  opposite  posi- 
tion, will  appear  from  the  trace  of  the  ink,  unless  the  line 
is  drawn  along  the  edge,  in  which  case  no  second  line  will 
appear.  The  edge  represents  the  straight  line  through 
the  two  points,  and  is  the  only  one. 

24.  A  second  fact  about  straight  lines  which  is  based 
directly  on  the  straight-line  axiom  is 

*  Two  different  straight  lines  can  intersect  in  but  one 
point. 

Whenever  a  statement  is  marked  with  the  mark  *,  that  statement 
requires  proof,  the  proof  in  the  first  part  of  the  Geometry  being  always 
closely  associated  with  the  definitions  and  axioms.  These  proofs  are 
sometimes  given  in  the  form  of  explanations,  but  the  pupil  should  in  all 
such  cases  be  able  to  explain  why  the  statement  is  true, 


26  RECTILINEAR  FIGURES 

This  is  the  first  example  of  contraposite  reasoning  in 
the  Geometry.  It  can  be  done  as  follows: 

If  the  two  lines  met  in  a  second  point,  there  would  be 
two  different  straight  lines  through  two  points. 

There  cannot  be  two  different  straight  lines  through 
two  points,  by  the  straight-line  axiom;  therefore,  by  con- 
traposite argument  (i.e.  the  conclusion  being  untrue,  the 
condition  is  also  untrue),  the  two  straight  lines  do  not 
meet  in  two  points. 

In  places  where  no  ambiguity  results,  "line"  will  be 
used  to  mean  straight  line,  since  when  any  other  kind  of 
line  is  meant,  the  kind  is  always  stated. 

25.  Sects.     A  limited  portion  of  a  line  is  called  a  sect, 
or  a  line  segment.     That  part  of  a  line  that  lies  between 
the  points  A  and   B  is  a  sect,          A  R 

and  is  called  AB.    Where  there    ' ' 

is  nothing  to  indicate  otherwise,  sect  will  mean  straight- 
line  sect. 

R  * 

26.  Broken  Lines.    A  line       ^^\/^^ — ^-^^ 
composed  of  sects  of  differ-     A  c  ^£ 
ent  lines  is  called  a  broken 

line^  as  ABODE.     A  broken  line  is  said  to  be  closed  if  it 

is  continuous;    that  is,  if   the  line,   when   traced   from 

any  point  through  its  entire 

length,  is  found  to  return 

to      the      starting      point. 

ABODE  is  not  a  closed  line, 

but  BSTUV  is  closed. 

The   straight-line   axiom 
shows  that  a  straight  line  is  not  continuous,  and  that  it 
does  not  intersect  itself, 


POINTS,   LINKS,   AND   SURFACES  27 

27.  Closed-line  Intersection  Axiom.     //  a  straight  line 
of  indefinite  length  passes  through  a  point  within  the 
surface  inclosed  by  any  closed  line,  it  intersects  the  closed 
line  at  least  twice. 

28.  Curved  Lines.     A  line, 
no  part  of  which  is  straight, 
is  called  a  curved  line. 

A  curved  line  also  is  called 
closed  if  it  is  continuous.     (See  §  26.) 

29.  Geometrical  Figures.     Any  combination  of  points, 
lines,  and  surfaces,  formed  under  given  conditions,  is  a 
geometrical  figure ;  as,  a  triangle  might  be  defined  as  the 
figure   formed  by  three  lines  meeting  in  pairs.     If   the 
figure  is  formed  by  straight  lines,  it  is  called  a  rectilinear 
figure. 

30.  Planes.     A  surface  in  which,  any  two  points  being 
taken,  the  straight  line  which  joins  them  lies  wholly  in 
the  surface  is  called  a  plane  surface,  or  simply  a  plane. 

A  plane  might  be  defined  as  the  surface  determined  by 
any  three  of  its  points  that  are  not  in  the  same  straight 
line. 

The  plane  occupies  the  same  position  among  surfaces 
that  the  straight  line  holds  among  lines.  As  indicated 
.by  the  first  definition  given,  it  is  straight  through  any  two 
of  its  points.  The  surface  of  a  blackboard  is  a  familiar 
example  of  a  plane. 

31.  Plane   Geometry.     Plane  Geometry  treats  only  of 
geometric  figures  that  lie  entirely  in  the  same  plane. 


SECTION   III.     EQUALITY 

32.  There  are  three  words  used  in  Geometry  to  denote 
equality  :  congruent,  equivalent,  and  equal. 

33.  Congruence.     Two  figures  are  congruent  when  they 
can  be  made  to  coincide  in  every  point. 

34.  Equivalence.     Closed  figures  (or  figures   bounded 
by  closed  lines,  see  §§  26,  28)  are  said  to  be  equivalent 
when  their  boundaries  inclose  the  same  amount  of  surface. 

35.  Equality.     The  word   equal  is  used  somewhat  in 
both  senses,  but  in  this  syllabus  it  will  be  used  only  in 
those  places  where  there  can  be  no  confusion  between  the 
ideas  of  congruence  and  equivalence.     For  example,  sects 
will  be  said  to  be  equal  when  they  can  be  made  to  coin- 
cide, even  though  this  fulfills  the  definition  of  congruence, 
for  since  a  sect  cannot  inclose  surface,  there  can  be  no 
confusion  with  equivalence. 

36.  Congruence  includes  equivalence,  whereas  equiva- 
lence does  not  imply  congruence ;  a  figure  inclosed  by  a' 
curved  line  might  be  equivalent  to  a  figure  inclosed  by  a 
broken  line  although  it  would  be  impossible  to  make  them 
coincide. 

37.  Geometric    Equality.      In   the    strictest   geometric 
sense,  equality  means  that  coincidence  is  possible,  and  in 
this  the  test  for  geometric  equality  differs  from  the  test 
for  arithmetic  equality,  for  two  arithmetic  magnitudes  are 
equal  if  they  contain  the  same  unit  the  same  number  of 

28 


EQUALITY  29 

times.  Evidently,  then,  equivalence  is  to  some  extent  an 
arithmetic  property,  but  Geometry  is  applied  so  often 
to  calculations  of  magnitudes  in  terms  of  a  unit,  that  it  is 
neither  necessary  nor  desirable  to  attempt  to  distinguish 
too  carefully  between  it  and  other  mathematical  subjects. 
In  practical  work,  Geometry  will  be  found  to  have  many 
parts  that  involve  Arithmetic  and  Algebra,  and  while  the 
distinctions  between  the  subjects  may  be  kept  in  mind, 
their  combined  use  is  entirely  legitimate. 

38.  Equality  Axioms. 

.  (1)  Things  equal  to  the  same  thing,  or  to  equal  things, 
are  equal  to  each  other. 

This  axiom  applies  to  both  congruence  and  to  equiva- 
lence ;  the  first  four  following  apply  only  to  sects,  angles 
(§  44),  and  equivalent  closed  figures ;  they  cannot  be  ap- 
plied to  the  congruence  of  closed  figures. 

(2)  If  equals  are  added  to  the  same  thing  or  to  equal 
things,  the  results  are  equal. 

(3)  If  equals  are  taken  from  the  same  thing  or  from 
equal  things,  the  results  are  equal. 

(4)  If  equals  are  multiplied  by  the  same  thing  or  by 
equal  things,  the  results  are  equal- 

(5)  If  equals  are  divided  by  the  same  thing  or  by  equal 
things,  the  results  are  equal. 

(6)  The  whole  equals  the  sum  of  all  its  parts. 

39.  Substitution   Axiom.       A  magnitude  may  be  put 
in  place  of  an  equal  magnitude  in  any  equation  or  state- 
ment of  inequality- 

This  axiom  is  not  independent  of  the  equality  axioms, 
but  it  is  more  convenient  for  many  purposes.  For  example, 
if  ak~b=r,  and  k  =  l;  then,  substituting  I  for  &,  al  —  b  =  r. 


30  RECTILINEAR   FIGURES 

40.  General  and   Geometric   Axioms.      Certain   of  the 
axioms  apply  not  only  to  geometric  magnitudes,  but  to  all 
magnitudes,  and  are  therefore  called  general  axioms.     The 
six  equality  axioms,  the  substitution  axiom,  and  the  axiom 
of  inequality  (§  82),  are  general  axioms.     The  straight-line 
axiom,  and  all  others  that  refer  to  geometric  conceptions 
only,  are  geometric  axioms. 

41.  Axiom  of  Motion.     Geometric  figures  can  be  moved 
about  in  space    without    altering    them    in    any  way. 
(Sometimes     stated,     "  without    altering    their   size  or 
shape-")     This  axiom  is  used  in  testing  congruence,  for 
one  figure  is  sometimes  supposed  to  have   been   placed 
upon  (or  superimposed  upon)  another  figure,  and  what- 
ever is  known  about  the  two  figures  is  then  used  to  deter- 
mine whether  or  not  they-  coincide. 

42.  Axiom  of  Division.    Any  magnitude  can  be  divided 
into  any  number  of  equal  parts.     (The  number  must  be 
a  positive  integer.) 

If  the  magnitude  is  divided  into  two  equal  parts,  it 
is  said  to  be  bisected;  if  into  three  equal  parts,  to  be 
trisected,  and  the  parts  are  called  halves,  and  thirds, 
respectively. 

*  43.    A  sect  can  be  bisected  by  but  one  point. 


If  P  and  Q  were  both  points  of  bisection  (or  midpoints) 
of  AB,  then  AP  and  AQ  would  be  equal,  since  each  is 
AB-t-2  (eq.  -*-  eq.).  Therefore  AP  and  AQ  coincide,  and  P 
falls  on  Q,  making  but  one  bisection  point. 

It  might  be  thought  at  first  that  this  fact  was  self-evi- 


EQUALITY  31 

dent,  but  many  things  can  be  bisected  in  different  ways, 
and  it  is  important  to  know  which  ones  have  but  one  pos- 
sible bisection.  An  apple,  for  instance,  could  be  cut  into 
halves  in  many  different  places. 

The  method  of  proof  used  to  show  that  a  sect  has  but 
one  midpoint  can  also  be  used  to  show  that  a  sect  has 
but  one  point  which  cuts  off  one  third  of  the  sect  from 
a  certain  end,  and  but  one  point  which  cuts  off  two 
thirds  from  that  end.  The  method  can  also  be  extended 
to  cover  all  cases  of  the  division  of  a  sect  into  equal  parts. 

NOTE.  A  method  of  proof  can  often  be  used  for  various  purposes 
with  very  little  change  except  for  minor  details.  It  has  already  been 
said  that  the  proof  in  §43  can  be  used  for  other  divisions  besides 
bisection.  It  is  also  true  that  it  can  be  used  for  the  bisection  of  an 
angle  as  well  as  of  a  sect,  and  for  any  other  divisions  of  an  angle. 
(See  §  64.)  It  is  very  important  that  a  student  should  become  famil- 
iar with  the  method  of  work  for  a  proof,  rather  than  to  attempt  to 
memorize  its  relatively  unimportant  details.  If  the  method  is  thor- 
oughly understood,  the  details  of  the  proof  are  not  likely  to  prove 
troublesome. 


SECTION   IV.     ANGLES 

44.    Angles.     If  two  lines  meet  in  a  point,  they  are  said 
to   form   an    angle.       The    common    point   is   called   the 
vertex,    and     the    lines    the 
arms  of  the  angle.     The  size 
of    the    angle    is     measured 
by    the   amount  of   rotation 
through  which  a  line  would 
pass  in  going  from  the  posi- 
tion of  one  arm  to  the  position  of  the  other  arm. 

In  the  accompanying  figure,  the  lines  AO  and  BO  form 
an  angle,  the  size  of  the  angle  being  measured  by  the 
amount  of  rotation  necessary  to  pass  from  the  position 
AO  to  the  position  BO. 

Since  there  are  two  ways  in  which  to  rotate  from  one 
line  to  the  other,  two  angles  are  formed  by  two  lines  diver- 
ging from  a  point;  when  it  is  necessary  to  distinguish 
between  the  two,  the  letters  on  the  arms  are  named  in 
the  order  of  direction  contrary  to  that  taken  by  the 
hands  of  a  clock,  the  vertex  letter  being  between  the 
other  two.  The  angle  marked  by  the  arrow  (which 
shows  the  correct  direction)  would  be  named  AOB,  while 
the  other  angle  would  be  BOA.  Ordinarily  the  smaller 
of  the  two  angles  is  meant,  such  an  angle  as  BOA  seldom 
being  considered  in  Geometry. 

If  there  is  little  chance  of  confusion  as  to  which  angle 
at  a  certain  vertex  is  being  used,  the  vertex  letter  alone 
is  often  used  to  name  the  angle ;  as,  angle  0. 

32 


ANGLES  33 

It  should  be  noticed  that  the  size  of  an  angle  in  no 
way  depends  on  the  length  of  its  "arms.  One  angle  would 
be  greater  than,  equal  to,  or  less  than,  another,  according 
as  it  would  include,  coincide  with,  or  fall  within,  that 
other,  if  it  were  placed  upon  it  with  the  vertex  and  one 
arm  coinciding.  An  angle  both  of  whose  arms  lie  between 
the  arms  of  a  second  angle  is  evidently  smaller. 

45.  Sign  for   "Angle."     The   sign  commonly  used  for 
"angle"  is  Z ;   as  Z  AOB.     If  there  is  a  chance  for  con- 
fusion with  the  signs  for  "greater"  and  "  less  "  (>  and  <), 
aline  is  sometimes  drawn  across  it,  as  ^.     The  commonest 
signs  and  abbreviations  will  be  found  following  the  index 
of  terms. 

46.  Adjacent  Angles.     If   two  angles   have  a  common 
vertex  and  lie  on  opposite  sides  of  a  common  arm,  they 
are    called    adjacent 

angles. 

/.AOB  and  Z  BOC 
are  adjacent  angles. 
They    are     adjacent 
whether   or   not   the   arms  AO  and   OC  lie  in  the   same 
straight  line. 

47.  Sum  and  Difference  of  Angles.     The  sum  of  two 

angles  is  the  angle  obtained  by  placing  the  angles  adjacent 
to  each  other,  and  ignoring  the  common  arm.  The  sum 
of  Z  AOB  and  Z  BOC  is  Z  AOC  (in  §  46). 

The  difference  of  two  angles  is  the  angle  obtained  by 
placing  the  angles  so  that  they  have  a  common  vertex, 
and  lie  on  the  same  side  of  the  common  arm.  The  dif- 
ference between  Z  AOC  and  Z  BOC  is  Z  AOB  (in  §  46). 

SMITH'S  SYL.  PL.  GEOM. — 3 


34 


RECTILINEAR   FIGURES 


48.  Vertical  Angles.     If  two  lines  intersect,  forming  four 
angles,  any  two  angles  that  are  not  adjacent  to  each  other 
are    called    opposite, 

or  vertical  angles. 

Z  A  OB  and  Z  COD 
are  vertical,  as  are 
Z  BOG  and  Z  DOA. 

The  arms  of  one  of.  two  vertical  angles  are  the  arms  of 
the  other  angle  extended  through  the  vertex. 

49.  Straight  Angles.     If  the  arms  of  an  angle  lie  in  the 
same  straight  line,  but  on  opposite  sides  of   the  vertex, 
the  angle  is  called  a 

straight  angle. 

Z  A  OB  is  a  straight       

angle.     The  sum  of 

all  the  successive  angles  around  a  point,  on  one  side  of 
a  straight  line,  is  equal  to  a  straight  angle;  for  example, 
Z  AOX+  Z  XOB  =  st.  Z. 

*50.  All  straight  angles  are  equal. 

For  the  arms  of  one  straight  angle  can  be  made  to 
coincide  with  the  arms  of  any  other  straight  angle,  since 
they  form,  in  each  case,  a  straight  line,  and  so  need  have 
but  two  points  in  common  to  coincide  throughout. 

51.   Supplements.    If  the  sum  of  two  angles  is  a  straight 
angle,  the  angles  are  called  supplements  of  each  other. 
In  §  49  Z.AOX  and  /.XOB  are  supplements. 

*52.  Supplements  of  the  same  angle,  or  of  equal  angles, 
are  equal  to  each  other. 


ANGLES  35 

For  if  equal  parts  are  taken  from  straight  angles  (which 
are  equal),  equal  parts  must  be  left,  by  equality  axiom  3. 
In  the  figure  :  st.  /.AOC  —  st.  ZXPZ. 

If  Z.AOB  =        Z  XPY, 
then  Z.BOC  =        Z  YPZ.  (eq.-eq.). 

NOTE.  When  one  wishes  to  refer  to  some  authority,  as  in  this 
case  to  the  equality  axiom,  any  clear  abbreviation  may  be  used.  The 
reference  (eq.  —  eq.)  means  "  equals  minus  equals,"  and  refers  to 
equality  axiom  3.  It  is  not  best  to  refer  to  authorities  by  number; 
all  authorities  should  be  quoted  in  such  a  way  that  one  could  readily 
understand  the  meaning  of  the  reference,  and  words  or  simple  ab- 
breviations are  usually  used. 

*53.   Any  two  vertical  angles  are  equal. 

For  they  are  supplements  of  the  same  angle. 

In  the  figure  of  §  48  Z  AOB  and  Z  COD  are  both  sup- 
plements of  Z  BOG  (or  of  ZDO^4);  similarly,  /.BOG  and 
Z  DO  A  are  both  supplements  of  Z  AOB  (or  of  Z  COD). 

54.  Perigons.  If  the  arms  of  an  angle  lie  in  the  same 
straight  line  and  on  the  same  side  of  the  vertex,  one  arm 
having  rotated  through  two  straight  angles,  the  angle  is 
called  a  perigon. 

Z  AOB  is  a  perigon,  since  OB  is  supposed  to  have  rotated 
around  O  (as  indicated  by  the  arrow)  to  the  position  OB 
on  OA. 

A  perigon  is  the  sum  of  the  successive  angles  around 
a  point;  and  it  equals 
two  straight  angles.  It 
is  in  the  sense  of  a  sum 
rather  than  as  a  single 
angle  that  perigon  is 
most  frequently  used  in  Geometry.  Z  AOX  +  Z  XOB  —  a 
perigon. 


36  RECTILINEAR  FIGURES 


*55.   All  perilous  are  equal. 

For  all  straight  angles  are  equal,  and  since  a  perigon  is 
twice  a  straight  angle,  perigons  are  equal  (eq.  x  eq.). 

56.  Explements.  If  the  sum  of  two  angles  is  a  perigon, 
the  angles  are  called  explements  of  each  other.  In  the 
figure  of  §54  /_AOX  and  Z.XOB  are  explements  of  each 
other. 

*57.  Explements  of  the  same  angle  or  of  equal  angles 
are  equal. 

Notice  the  similarity 
to  §  52. 

58.   Right  Angles.  If 

one  line  meets  another 

line  so  as  to  make  the 

two     adjacent     angles 

equal,    the   angles   are 

called  right  angles,  and 

the   lines   are   said   to   be  perpendicular   to   each   other. 

If  /.AOB  =  /-BOC,  then  these  angles   are   right   angles. 

A  line  not  perpendicular  to  a  second  line  is  said  to  be 

an  oblique  to  that  line. 

*59.   A  right  angle  is  one  half  a  straight  angle. 
For  by  the  definition  of  right  angle,  a  straight  angle  is 
the  sum  of  two  equal  right  angles. 

*60.   All  right  angles  are  equal. 

For  they  are  halves  of  equal  straight  angles.  What 
axiom  is  used?  • 

61.  Complements.  If  the  sum  of  two  angles  is  a  right 
angle,  the  angles  are  called  complements  of  each  other.  In 
§  58  Z  AOX  and  /.XOB  are  complements  of  each  other. 


ANGLES 


37 


*62.  Complements  of  the  same  angle  or  of  equal  angles 
are  equal. 

See  §§  52  and  57. 

63.  Acute,  Obtuse,  Reflex  Angles.  An  angle  less  than 
a  right  angle  is  called  acute;  one  greater  than  a  right 
angle,  but  less  than  a  straight  angle,  is  called  obtuse  ;  one 
more  than  a  straight  angle,  but  less  than  a  perigon,  is 
called  reflex. 

*64.   An  angle  can  be  bisected  by  but  one  line. 
If    OX  and  OF   both 
bisect      Z   AOB,      then 


Z  AOB  -^  2,  and  eq.  -*- 
eq.).  Therefore  the 
angles  coincide,  and  OX 
falls  on  OF;  that  is, 
there  is  but  one  line 
which  bisects  the  angle. 

*65.  At  a  given  point  in  a  given  line  there  can  be  but 
one  perpendicular  in  the  same  plane. 

For  the  perpendicular  bisects  the  straight  angle  having 
its  vertex  at  its  foot  (§  59),  and  there  can  be  but  one 
bisector  of  an  angle  (§64). 

*66.  The  bisectors  of  two  vertical  angles  lie  in  one 
straight  line. 

ZA'OF  is  one  half  the  perigon,  or  a  straight  angle,  for 
the  Zs  XOB,  BOC, 
COF,  are  respec- 
tively equal  to  the 
Zs  AOX,  DO  A,  YOD. 
Therefore  Z.YOF  is 
a  straight  angle,  and  XY  is  a  straight  line. 


38  RECTILINEAR   FIGURES 

*67.  The  bisectors  of  two  adjacent  supplements  are 
perpendicular  to  each  other. 

For  the  angle  between  them  is  one  half  the  straight 
angle. 

68.  Axiom  of  Intersection.  An  indefinite  line  drawn 
from  the  vertex*  of  an  angle  that  is  less  than  a  straight 
angle,  between  the  arms  of  the  angle,  intersects  any  line 
that  joins  a  point  on  one  arm  of  the  angle  with  a  point 
on  the  other  arm  of  the  angle. 

For    example,     the 

line  OK  meets  ES,  or  s. 

any  other  line  that 
joins  any  point  on  OX 
with  a  point  on  OF. 

A     consequence    of 
this     axiom     that    is 
needed  for  some  of  the  proofs  will  be  found  in  the  Appen- 
dix, §  343. 


SECTION   V.     POLYGONS 


69.  Polygons.     The  limited  portion  of  a  plane  bounded 
by  a  broken  line  which  is  closed  (§  26)  is  called  a  polygon. 
The  amount  of  surface  within  the  polygon  is  called  the 
area  of  the  polygon,  and  the  length  of  the  boundary  is  called 
the  perimeter  of  the   polygon.     The   term    "perimeter" 
is  often  used  for  the  broken  line  bounding  the  polygon 
when  no  idea  of  length  is  involved,  but  when  this  is  done, 
the  context  usually  makes  clear  which  meaning  is  to  be 
attached  to  the  word. 

70.  Vertices   and   Sides.     The   vertices   of  the   angles 
formed  by  the   sects  of  the  broken  line  are  called  the 
vertices  of  the  polygon,  and  the  sects  themselves  are  called 
the  sides  of  the  polygon.     The  figure  ABODE  is  a  polygon. 

71.  Angles    of   a 
Polygon.     The  angles 
on  the  left  in  passing 
around  a  polygon  in 
the  direction  contrary 
to  that  taken  by  the 
hands  of  a  clock  are 
called  interior  angles. 
The  angles  of  ABODE 

which  are  marked  with  the  arrows  are  interior  angles. 

The  angle  at  any  vertex  of  a  polygon  having  as  arms 
one  side  of  the  polygon,  and  the  continuation  of  another 
side,  is  called  an  exterior  angle,  as  Z  XBC.  In  speaking 

39 


X 


40  RECTILINEAR  FIGURES 

of  the  exterior  angles  of  a  polygon,  the  angles  formed  by 
producing  the  sides  in  succession,  each  through  the  vertex 
formed  with  the  following  one,  in  passing  around  the 
polygon,  are  meant.  There  are  evidently  two  sets  of 
exterior  angles  of  a  polygon,  formed  by  passing  around 
clockwise  or  counterclockwise.  These  sets  are,  however, 
equal,  for  the  two  exterior  angles  at  any  vertex  are  ver- 
tical. 

72.  Diagonals.     A    line  joinihg   two    non-consecutive 
vertices  of  a  polygon  is  called  a  diagonal. 

73.  Concave,  Convex,  and  Cross  Polygons.     A  polygon 
is  said  to  be  convex  if  no  side  when  produced  could  cut  the 
surface  of  the  polygon.     Unless  otherwise  stated,  convex 
polygon  will  be  meant  whenever  the  term  "  polygon  "  is 
used. 

A  polygon  is  concave  when  at  least  one  side,  if  produced, 
would  cut  its  surface  ;  it  is  called  cross  when  its  perimeter 
intersects  itself. 

74.  Equilateral,  Equiangular,  and  Regular  Polygons.     A 

polygon  that  has  all  its  sides  equal  is  called  equilateral; 
one  that  has  all  its  angles  equal  is  called  equiangular.  A 
polygon  that  is  both  equilateral  and  equiangular  is  called 
regular. 

75.  Number  of  Sides.     A  polygon  of  three,  four,  five, 
six,  eight,  ten  sides  is  called,  respectively,  a  triangle,  quad- 
rilateral, pentagon,  hexagon,  octagon,  decagon,  etc. 

76.  Base.     The  side  of  a  polygon  on  which  it  appears 
to  stand  is  called  its  base.     Any  side  of  a  polygon  might 
be  considered  the  base. 


POLYGONS  41 

TRIANGLES 

77.  Parts  of  a  Triangle.     A  triangle  has  six  parts,  three 
sides  and  three  angles.     An  angle  and  a  side  are  spoken 
of  as  opposite  to  each  other  when  the  side  'is  not  one  of 
the  arms  of  the  angle.     A  side  is  sometimes  spoken  of  as 
included  between  two  angles,  and  an  angle  as  included 
between  two  sides,  when  the  order  in  which  they  lie  is 
meant. 

78.  Vertex  Angle.     The  angle  opposite  the  base  of  a 
triangle  is  called  the  vertex  angle,  or  the  vertical  angle. 

79.  Equality  of   Sides.     If   a   triangle  has  two   equal 
sides,  it  is  called  isosceles ;  if  no  equal  sides,  scalene.     In 
an  isosceles  triangle,  the  equal  sides  are  sometimes  called 
legs,  the  third  side  the  base. 

80.  Angles    of    a   Triangle.     If    all    the   angles    of    a 
triangle  are  acute,  it  is  called  an  acute-angled  triangle;  if 
one  angle  is  right,  it  is  called  a  right  triangle  ;  and  if  one 
angle  is  obtuse,  it  is  called  an  obtuse- angled  triangle.     In  a 
right  triangle  the  side  opposite  the  right  angle  is  called  the 
hypotenuse,  and  the  other  sides  the  legs. 

81.  Lines  of  a  Triangle.     There  are  four  kinds  of  lines 
of  importance  in  work  with  a  triangle:  the  bisectors  of  the 
angles,  the  perpendicular  bisectors  of  the  sides,  the  altitudes, 
and  the  medians.     The  first  two  explain  themselves  ;  an 
altitude  is  a  perpendicular  from  a  vertex  to  the  opposite 
side,  and  a  median  is  a  line  from  a  vertex  to  the  midpoint 
of  the  opposite  side.    If  the  altitude  of  a  triangle  is  spoken 
of,  the  altitude  to  the  base  is  meant. 


SECTION   VI.     INEQUALITIES 

82.  Axiom'  of    Unequals.     The  whole  is  greater  than 
any  of  its  parts. 

83.  Inequalities.     There  are  certain  truths  relating  to 
statements  of  inequality  that  depend  very  closely  on  the 
general  axioms.     Their  proofs  are  given  under  the  head  of 
Inequalities  in  nearly  all  Algebras,  so  they  will  not  be  con- 
sidered here.     They  ^  are  used  for  those   magnitudes  for 
which  equality  axioms  2-5  are  used. 

(1)  If  equals  are  added  to,  taken  from,  multiplied  ~by, 
or  divided  into,  unequals,  the  results  are  unequal  in  the 
same  sense. 

(That  is,  the  greater  quantity  remains  greater  after  the 
operation  is  performed.) 

(2)  If  unequals  are  taken  from,  or  divided  into  equals, 
the  results  are  equal  in  the  opposite  sense. 

(3)  If  unequals  are  added  to,  or  multiplied  by,  un- 
equals in  the  same  sense,  the  results  are  unequal  in  the 
same  sense. 

(4)  If  the  first  of  several  magnitudes  is  greater  than 
the  second,  the  second  greater  than  the  third,  the  third 
greater  than  the  fourth,  and  so  on,  then  the  first  is  greater, 
than  the  last. 

NOTK.  (-t)  holds  for  "the  first  less  than  the  second,"  etc.;  also 
any  pairs  might  be  equal  without  changing  the  result,  if  there  is  at 
least  one  inequality. 

These  statements  are  all  in  regard  to  positive  magni- 
tudes ;  they  are  not  all  true  when  negative  quantities  are 
used. 

WARNING.  Unequals  should  not  be  taken  from,  or  divided  into,  un- 
euuals,  for  the  results  cannot,  in  general,  be  determined. 

43 


SECTION  VII.    PROPOSITIONS 

84.  Propositions.     Proposition  is  a  general  term  includ- 
ing: 

(1)  Theorem,  which  is  a  truth  to  be  proved. 

(2)  Corollary,  which  is  also  a  truth  to  be  proved,  but 
generally  one  that  follows  quite  directly,  and  often  very 
simply,  from  a  known  truth  —  most  often  from  a  theorem 
that  has  just  been  proved. 

(3)  Problem,  or  Construction  Theorem,  which  requires 
that  a  certain  figure   be   drawn  from    given   parts.       A 
more  definite  understanding  of  problem  will  be  given  in 

§  I??- 

85.  Parts  of  a  Theorem.     A  theorem  is  composed  of  the 
condition,  or  hypothesis,  and  the  conclusion.       It   usually 
takes  the  form,  —  If  a  certain  condition  is  true,  a  certain 
conclusion  is  also  true.     (See  §  2.) 

86.  Proof  of  a  Proposition.     To  prove  any  proposition, 
the  student  has  certain  materials  from  which  to  work, 
namely  :  the  foregoing  definitions,  axioms,  and  truths  con- 
cerning them ;    the  theorems  preceding  the  one   that  is 
being   proved ;    and  the   condition  of  the  proposition  in 
question.     From  these  known  truths  the  proof  must  be 
deduced,  and  the  required  conclusion  must  be  reached. 

The  proof  must  always  be  a  logical  one ;  the  truth  of  a 
proposition  must  not  be  judged  by  measurements  —  as  in 
Concrete  Geometry  —  or  from  the  appearance  of  the  figure, 
for  such  methods  have  no  place  in  this  subject.  It  might 
be  said,  however,  that  a  carefully  drawn  figure  will  some- 

43 


44  RECTILINEAR   FIGURES 

times  give  the  idea  which  suggests  the  correct  proof,  al- 
though the  appearance  of  the  figure  cannot  be  quoted  as 
authority  for  the  truth  of  any  statement. 

87.  Order  of  Proof.     The  following  order  of  proof  has 
been  found  very  convenient,  and  the  student  is  advised  to 
follow  it  in  all  work. 

STATEMENT  OF  THEOREM 
Figure 

Given.      Condition,  in  terms  of  the  letters  of  the  figure. 

To  prove.  Conclusion,  in  terms  of  the  letters  of  the 
figure. 

Proof.  I.  The  proof,  in  numbered  steps,  with  the  author- 
ity for  each  step  following  it. 

NOTES.  Any  special  case  or  interesting  fact  concerning  the 
theorem. 

COROLLARIES.     Those  connected  with  the  theorem. 
Examples  of  this  form  will  be  found  in  the  proofs  given 
later. 

88.  Classification  of  Theorems.     All  theorems  of  Plane 
Geometry  may  be  divided  into  certain  groups  which  might 
be  called  classes  ;  for  example,  some  theorems  prove  angles 
equal,  others  prove  lines  unequal,  still  others  prove  tri- 
angles congruent,  etc. 

It  is  evident  that  any  theorem  of  a  certain  class  must 
depend  upon  something  which  will  prove  the  particular  re- 
sult desired  ;  that  is,  upon  something  in  its  own  class,  —  unless 
it  can  be  obtained  by  logic  from,  another  class.  Except  in 
the  cases  of  logic,  which  are  readily  recognized,  each  geo- 
metric truth  depends  directly  upon  some  preceding  truth 
of  its  own  class,  and  the  foundation  truths  of  each  class  are 
the  definitions  and  axioms  that  concern  the  things  used. 


CLASSIFICATIONS  45 

89.    Important  Classes  Already  Started. 

(1)  Congruence.       To    prove   figures    congruent,    they 
must  be  shown  to  coincide ;  the  axiom  of  motion  can  be 
used  to  suppose  one  of  two  figures  placed  on  the  other  in 
order  to  test  the  coincidence. 

(2)  Angles  Equal.     Right  angles,  straight  angles,  peri- 
gons,  complements  of  equal  angles,  supplements  of  equal 
angles,  explements  of  equal  angles,  vertical  angles. 

(3)  Magnitudes  Unequal.     The  inequality  axiom :  after 
having  obtained  one  inequality,  the  inequality  statements 
can  be  used.     It  follows  from  this  that  the  only  way  to 
prove  two  things  unequal  at  this  stage  of  the  work  is  to 
show  that  one  is  a  part  of  the  other,  or  that  one  equals  a 
part  of  the  other. 

(4)  Line  Straight.     A  line  is  straight  if  its  sects  are 
the  arms  of   a   straight  angle ;    the  bisectors  of   vertical 
angles  lie  in  a  straight  line. 

(5)  Lines  Perpendicular.     If  the  adjacent  angles  formed 
by  them  are  equal ;    if  they  bisect  adjacent  supplemental 
angles. 

(6)  Lines  Coincide  (or  are  determined).      If  they  go 
through  the  same  two  points,  if  they  bisect  the  same  angle, 
if  they  are  perpendicular  to  the  same  line  at   the  same 
point  (in  a  plane). 

(7)  Points  Coincide  (or  are  determined).     If  they  are 
intersections  of  the  same  two  lines,  if  they  are  midpoints 
of  the  same  sect,  if  they  are  correspondingly  placed  points 
of  equal  sects  which  are  made  to  coincide. 

(8)  Lines  Equal.     This  would  be  a  case  of  congruence ; 
the  lines  would  have  to  be  shown  to  coincide. 

These  are,  of  course,  not  all  the  classes  to  be  found  in 
Plane  Geometry,  but  they  serve  to  show  that  a  founda- 


46  RECTILINEAR   FIGURES 

tion  has  already  been  laid,  upon  which  certain  classes  of 
geometric  truths  can  be  established. 

90.  Method  of  Attack. 

(1)  Determine  the  class  of  the  theorem. 

(2)  List  the  known  ways  of  proving  the  required  con- 
clusion   (or,  in   other   words,  list   the   ones  of  the  same 
class). 

(3)  Examine  each  way  found  as  to  the  possibility  of  its 
being  applied  to  the  figure  in  question,  and  especially  to  ' 
conditions  of  the  theorem. 

(4)  Having  decided  on  one  or  more  ways  as  possibili- 
ties (probably  by  eliminating  those  which  seem  impossible 
of  application),  try  to  reason  from  the  condition  to  the 
conclusion  by  the  method  chosen. 

WARNING.  Remember  that  it  is  necessary  to  obtain  the  conclusion, 
and  to  use  the  condition;  more  trouble  is  made  by  neglect  to  think  of  these 
two  points  than  by  anything  else  in  Geometry. 

91.  ORAL  AND  REVIEW  QUESTIONS 

Define  contraposite,  straight  line,  adjacent  angles,  converse,  ver- 
tical angles,  axiom,  perigon,  right  angle,  complements,  theorem. 
Are  two  supplemental  angles  ever  equal?  always  equal?  What  is 
the  complement  of  one  half  a  right  angle?  of  a  right  angle?  of  a 
zero  angle  ?  of  three  quarters  of  a  right  angle  ?  What  is  the  sup- 
plement of  a  right  angle?  of  a  right  angle  and  a  half?  of  two  right 
angles?  What  does  a  perpendicular  do  to  a  straight  angle?  Why 
are  all  straight  angles  equal?  all  right  angles?  all  perigons  ?  How 
many  points  are  needed  to  fix  a  line?  How  many  lines  are  needed 
to  fix  a  point?  How  many  bisecting  points  can  a  sect  have?  Why? 
How  many  bisecting  lines  can  an  angle  have?  Why?  How  does 
the  angle  case  apply  to  perpendiculars?  What  method  of  proof 
applies  to  proving  that  complements,  supplements,  and  explements 
of  equal  angles  are  equal?  Apply  one  of  these  to  show  that  vertical 
angles  are  always  equal.  State  the  equality  axioms ;  the  straight- 
line  axiom;  the  di\7ision  axiom;  the  substitution  axiom;  the  inter- 


ORAL   AND   REVIEW   QUESTIONS  47 

section  axiom.  Upon  what  foundation  must  the  proofs  of  the  propo- 
sitions be  based?.  If  a  statement  is  known  to  be  true,  what  related 
statement  is  always  true?  What  statement  is  sometimes  true,  and 
when?  State  the  obverse  of  "This  statement  is  true."  Of  what  is 
a  straight  angle  the  sum?  a  perigon?  Of  what  two  equal  angles  is  a 
straight  angle  the  sum  ?  a  perigon  ?  Of  what  four  equal  angles  is  a 
perigon  the  sum?  If  an  angle  is  twice  its  complement,  how  large  is 
the  angle  ?  If  an  angle  is  three  times  its  explement,  how  large  is  it? 
Upon  what  does  the  size  of  an  angle  depend  ?  What  ways  are  known 
to  prove  figures  congruent?  angles  equal?  angles  unequal?  lines 
equal?  lines  unequal ?  lines  perpendicular?  lines  not  perpendicular? 
How  can  a  line  be  determined?  a  point  be  determined?  What  order 
of  proof  can  be  used  for  a  theorem?  Define  each  of  the  parts  of  the 
proof.  Distinguish  between  the  terms  "  proposition,"  "  theorem," 
"  problem." 


SECTION   VIII.    TRIANGLE  THEOREMS 

92.  Theorem  I.  //  two  triangles  have  two  sides  and 
the  included  angle  of  the  one  respectively  equal  to  two 
sides  and  the  included  angle  of  the  other,  the  triangles 
are  congruent. 

ANALYSIS 

Class.     Triangles  congruent. 

Known  Methods.  Coincidence,  using  the  axiom  of 
motion. 

Method  to  be  used.  One  triangle  will  be  supposed  to 
be  placed  on  the  other,  and  the  given  facts  will  then  be 
used  to  determine  whether  they  would  coincide. 

PROOF 


Given.     A  ABC,  A  DEF ;  BC  =  EF,  CA  =  FD,  Z  c  =  ^  F. 

To  prove.    A  ABC  ^  A  DEF. 

Proof.     I.    Suppose  A  ABC  to  be  placed  on  A  DEF,  C  on 

F,  and  BC  along  EF  (ax.  of  motion). 
Then  CA  would  lie  along  FD,  vZc=^F 
(given). 

48 


TRIANGLE   THEOREMS  49 

II.    B  -would  fall  on  E,-.-BC=EF  (given). 
A  would  fall  on  D,  •.•  CA  =  FD  (given). 

III.  AB  would  coincide  with  DE  (but  one  straight 

line  through  two  points). 

IV.  .-.  &ABC  ^  ADEF  (def.  ^). 

NOTE.  It  often  makes  the  conditions  of  the  theorem  more  clear  if 
those  conditions  are  indicated  in  the  figure.  The  usual  way  of  show- 
ing equal  parts  is  to  place  a  like  mark  on  any  two  parts  that  are 
known  to  be  equal.  In  the  figure  used  in  Th.  I,  the  equal  parts  are 
indicated  by  such  marks.  Where  equal  parts  are  used  in  the  theorem, 
although  not  given,  the  same  method  is  sometimes  used. 

93.  Corresponding  Parts  of  Congruent  Figures.     When 
two   figures   coincide,  each  part    (side  or  angle)  of   one 
coincides  with  a  part  of  the  other,  and  is  therefore  equal 
to  it.     Two  parts  of  congruent  figures  that  would  coin- 
cide if  the  figures  were  made  to  coincide  are  called  corre- 
sponding, or  homologous  parts. 

When  two  figures  are  known  to  be  congruent  on  account 
of  their  having  certain  equal  parts, —  as  by  Th.  I,  —  the 
other  corresponding  parts  can  be  told  by  their  position 
relative  to  the  known  parts;  as,  by  their  being  opposite 
to  known  parts,  or  between  two  known  parts.  In  Th.  I, 
AB  =  DE,  Z  ^t  =  Z  D,  Z  #  =  Z  j£. 

The  most  important  use  of  congruence  of  figures  is  to  prove 
equality  of  lines  and  of  angles. 

94.  Theorem  II.   If  two  triangles  have  two  angles  and 
the  included  side  of  the  one  equal  to  two  angles  and  the 
included  side  of  the  other,  the  triangles  are  congruent. 

ANALYSIS 

This  theorem  is  of  the  same  class  as  Th.  I,  so  must  be 
done  in  the  same  way,  or  else  by  means  of  Th.  I.     It  can 
be  proved  in  either  of  these  ways,  and  the  superposition 
SMITH'S  SYL.  PL.  GBOM. — 4 


50  RECTILINEAR  FIGURES 

method  is  much  the  easier.     It  is  so  much  like  Th.  I  that 
the  proof  will  be  left  for  the  student  to  do  for  himself. 

EXERCISES 

1.  Prove  Th.  II  by  means  of  Th.  I. 

HINT.  A  second  side  is  all  that  is  needed.  Suppose  that  the  second 
side  of  one  is  not  equal  to  the  corresponding  side  of  the  other,  and  cut  off 
a  sect  on  it  which  is  equal  to  the  corresponding  side  of  the  other,  thus  farm- 
ing a  triangle  which  is  congruent  to  the  second  triangle.  Now  examine  the 
angles  in  the  Jig  tire. 

2.  A  point  in  the  perpendicular  bisector  of  a  sect  is  equidistant 
from  the  ends  of  the  sect. 

3.  In  an  equilateral  triangle,  the  bisector  of  any  angle  forms  two 
congruent  triangles. 

4.  If  the  diagonals  of  a  quadrilateral  bisect  each  other,  the  quad- 
rilateral has  two  pairs  of  equal  sides. 

5.  If  a  diagonal  of  a  quadrilateral  bisects  two  angles,  the  quadri- 
lateral has  two  pairs  of  equal  sides. 

WARNING.  In  exercises  4  an(l  £>  be  certain  that  the  sides  used  are 
corresponding  parts  of  the  triangles  found. 

6.  In  a  regular  pentagon,  the  diagonals  are  equal. 

7.  If  a  line  is  drawn  from  the  end  of  a  line  sect  to  a  point  in  its 
perpendicular  bisector,  the  line  from  the  other  end  of  the  sect  making 
the  same  angle  with  the  sect  will  meet  the  perpendicular  bisector  at 
the  same  point. 

8.  If  two  triangles  are  congruent,  medians  drawn  to  corresponding 
sides  are  equal. 

9.  If  the  bisector  of  an  angle  of  a  triangle  is  perpendicular  to  the 
opposite  side,  the  triangle  is  isosceles. 

10.  Two  equal  lines  AC  and  AD  are  drawn  on  opposite  sides  of 
AB,  making  equal  angles  with  AB.  Prove  that  BC  and  BD  will  also 
make  equal  angles  with  AB. 


TRIANGLE   THEOREMS  51 

11.  If  two  triangles  are  congruent,  the  bisectors  of  corresponding 
angles  are  equal. 

12.  If  one  line  is  perpendicular  to  a  second  line,  then  two  lines 
drawn  from  a  point  in  the  first  line  to  the  second  line  are  equal  if  they 
make  equal  angles  with  the  first  line. 

NOTE.  It  was  said  in  §  93  that  the  most  important  use  of  con- 
gruent figures  was  to  prove  lines  equal  and  angles  equal.  It  is  often 
necessary  to  choose  between  different  congruent  theorems,  and  the  re- 
sult desired  affects  greatly  the  choice  of  the  theorem. 

For  example,  if  lines  are  to  be  proved  equal,  one  is  much  more  likely 
to  be  able  to  use  Th.  II  than  Th.  I,  for  it  requires  less  knowledge 
about  equal  lines  —  the  thing  that  is  being  worked  for.  So,  if  angles 
are  to  be  proved  equal,  Th.  I  is  better  than  Th.  II,  for  one  is  more 
likely  to  know  about  the  lines  than  about  the  angles  when  the  de- 
sired end  is  equality  of  angles. 

Such  considerations  as  the  above  are  often  of  use  in  finding  the 
proof  of  a  theorem,  and  the  pupil  should  give  the  question  under 
examination  very  careful  thought  before  attempting  to  write  the 
proof. 

95.  Theorem  III.  If  a  side  of  a  triangle  is  extended, 
the  exterior  angle  formed  is  greater  than 

(1)  the  angle  opposite  the  side  extended; 

(2)  either  angle  not  adjacent. 
[Note  that  (2)  includes  (1).] 

NOTE.  This  theorem  .is  a  hard  one  to  discover  at  the  beginning 
of  Geometry.  It  is,  however,  a  very  useful  theorem,  and  a  very  com- 
plete example  of  the  classification  method  of  finding  a  construction 
and  proof.  The  pupil  need  not  feel  discouraged  by  its  apparent  diffi- 
culty, but  should  rather  regard  this  analysis  as  an  example  of  the 
method  to  be  applied  to  the  following  theorems,  most  of  which  are 
much  less  difficult- 

The  proof  of  this  theorem  will  be  easy  to  understand  after  the 
auxiliary  lines  (or  extra  lines  which  must  be  added  to  the  figure  in 
order  to  obtain  the  proof)  are  found.  Notice  that  the  classification 
method  shows  what  additional  lines  are  needed  in  the  figure,  when  the 
given  figure  does  not  itself  include  all  the  material  necessary. 


52  RECTILINEAR  FIGURES 

ANALYSIS.     (PART  I) 

Class.     Angles  unequal. 

Preceding  Methods.     Inequality  axiom. 

Application. 


It  is  necessary  to  show  that  Z  XBC  >  Z  C ;  that  is, 
that  Z  C  is  part  of  Z  XBC,  which  is  evidently  untrue,  or 
that  Z  C  is  equal  to  part  of  Z.XBC.  Since  this  requires 
angles  equal,  a  new  class  must  be  used. 

Class.     Angles  equal. 

Preceding  Methods.  Those  in  the  list  in  §  89,  (2)  ;  cor- 
responding parts  of  congruent  triangles. 

Elimination.  Z  C  is  not  known  to  be  a  rt.  Z,  st.  Z,  or 
a  perigon ;  there  are  no  equal  angles  of  which  Z  C  and 
part  of  Z  XBC  could  be  complements,  supplements,  or  ex- 
piemen  ts;  it  is  not  vertical  to  any  angle  at  B.  There 
remains  only  the  possibility  that  Z  C  and  an  angle  at  B 
may  be  corresponding  angles  of  congruent  triangles. 

This  requires  still  a  third  classification. 

Class.  Triangles  congruent  (triangles,  because  no  ap- 
plicable way  of  proving  other  figures  congruent  is  known 
as  yet). 

Preceding  Methods.  2  s.  incl.  Z  ;  2  A  incl.  s.  Prob- 
ably 2  s.  incl.  Z  will  be  used  because  angles  are  to  be 
proved  equal. 


TRIANGLE   THEOREMS  58 

Application.     Any  line  meeting  BC  and  CA,  as  RS,  will 
form  a  triangle  of  which  Z  <7  is  a  part,  and  any  line  from 


B  to  the  extension  of  RS,  as  BK,  will  form  a  triangle  con- 
taining a  part  of  Z  XBC.  It  remains  only  to  draw  these 
lines  in  such  a  way  that  the  triangles  formed  will  be  con- 
gruent. Since  Z  C  is  to  be  corresponding  to  Z  KBS,  the 
opposite  sides  must  be  made  equal,  and  this  can  be  done  by 
cutting  off  SK=  RS. 

But  Z.C8E  =  Z  KSB  (vert.),  and  the  only  other  lines 
needed  are  CS  and  SB,  for  these,  with  the  parts  named, 
would  make  two  sides  and  the  included  angle.  As  RS  can 
be  drawn  anywhere,  let  8  be  the  midpoint  of  BC. 

Therefore,  to  make  the  triangles  congruent,  but  two  things 
are  needed :  That  a  line  be  drawn  from  any  point  on  CA 
through  the  midpoint  of  BC,  and  extended  its  own  length,  the 
extremity  being  joined  to  B. 

The  proof  has  now  been  discovered  if  Z  KB 8  is  a  part  of 
Z  XBC.  It  is  a  part  of  it  if  it  lies  within  Z  XBC-,  that  is, 
if  the  point  K  is  between  BX  and  BC.  K  does  lie  between 
these  arms  unless  line  RSK  meets  one  arm  of  /.XBC  after 
intersecting  BC  at  8.  It  cannot  meet  BC  again  (two  st. 
lines  meet  in  but  one  point),  and  it  cannot  meet  BX  again 
if  it  has  already  met  it.  The  simplest  way  to  have  it  meet 


54 


RECTILINEAR  FIGURES 


BX  and  still  be  drawn  from  a  point  on  CA  is  to  draw  it 
from  A.     (See  §  96.) 

PROOF 


Given.     A  ABC-,  AB  extended  to  X. 
To  prove.     (1)  Zx£C>Zc;   (2) 
Proof.     I.    Draw  AK  from  A,  through  M,  the  midpoint 
of  BC,  to  -ET,  so  that  MK  =  AM.     Draw  BK. 
II.    CM  =  MB  (bisection)  . 
Z  CM  A  =  Z  KMB  (vert.). 
AM  =  MK  (const.). 

.-.  ACMA  ^AKMB  (2s.  incl.  Z). 

III.  .-.  Z  C  =  Z  KBM  (cor.  pts.). 

IV.  But  Z  KBM  is  part  of  Z  XBC  (AMK  having 

met  both  arms). 
.-.Z.XBOZ.KBM  (ineq.  ax.). 
V.    .  •  .  Z  XBC  >  Z  C  (sub.  ax.). 

NOTE.     This  proves  an  exterior  angle  greater  than  the  interior 
angle  opposite  the  side  produced. 

VI. 


VII. 


(2)  Extend  CB  to    F;     then 

since  Z  A  is  opposite  the   extended  side 

CB  (part  1). 

But  ZCBX=  /.ABY  (vert.). 
.-.  /.CBX>/.A.  (sub.  ax.). 


TRIANGLE   THEOREMS  55 

96.  Location.     That  part  of  Th.  Ill  which  shows  that 
point  K  lies  between  the  arras  of  /.  XBC,  and  all  other 
parts  of  the  Geometry  where  the  place  in  which  a  point 
or  line  must  lie  is  considered,  are  said  to  deal  with  the 
location  of  the  thing  in  question.     It  is  very  important 
that  the  place  of  each  point  or  line  in  the  figure  be  fixed 
absolutely,  as   otherwise   incorrect   proofs  will  often   be 
given.     (See  §  110.) 

97.  COR.  1.    There  can  be  but  one  perpendicular  from, 
a  given  point  to  a  given  line. 

Assume  one  perpendicular,  and  show  that  any  other  line 
from  the  same  point  will  make  an  obtuse  angle. 

98.  Determination  of  Lines.     §  97  is  another  way  to  de- 
termine, or  fix  a  line.       Three  ways  have  already  been 
mentioned  in  §  89  (6)  ;  they  are,  by  two  points,  by  its  bi- 
secting an  angle,  by  its  being  perpendicular  to  a  line  at  a 
point  in  the  line.     The  new  one  is  by  its  being  perpendic- 
ular to  a  line  from  a  point  outside  the  line.     The  last  two 
are  usually  spoken  of  as  perpendicular  to  a  line  at  a  point 
and  from  a  point. 

In  making  a  construction  line  to  help  in  obtaining  a 
proof,  the  line  can  be  drawn  so  as  to  do  any  one  of  these 
four  things,  but  no  more  than  one.  For  example,  in  Th. 
Ill,  the  line  AM  is  drawn,  and  it  is  determined  by  A 
and  3f;  then  it  is  extended  to  K,  so  that  MK  =  AM.  This 
extension  simply  represents  another  part  of  the  line  de- 
termined by  A  and  3f,  of  which  AM  is  one  sect.  The  line 
SB  is  determined  by  the  points  K  and  B. 

It  should  be  noticed  that  the  determination  of  a  line 
usually  depends  upon  the  determination  of  one  or  more 
points  ;  as  in  the  case  just  discussed,  the  point  A  is  given, 
and  the  point  M  is  determined,  because  there  is  but  one 


56  RECTILINEAR   FIGURES 

midpoint  in  a  sect.     K  is  determined  because  the  sect  MK 
equals  the  sect  AM. 

WARNING.  Never  attempt  to  make  a  line  do  two  determining  things 
at  the  same  time  ;  as,  bisect  an  angle  and  go  from  the  vertex  to  a  fixed 
point ;  or  join  tico  known  points  and  be  perpendicular  to  a  Jixed  line. 
Always  determine  each  line,  but  by  one  only  of  the  determining  conditions. 

The  lines  discussed  in  this  paragraph  — auxiliary  lines 
—  are  only  representations  of  lines  that  do  exist.  Each 
point  or  line  added  to  a  figure  is  the  representation  of 
some  actual  point  or  line  that  can  exist  in  the  figure, 
therefore  it  is  necessary  to  be  especially  careful  not  to 
attempt  to  draw  lines  that  are  not  possible. 

99.  COR.  2.    If  a  triangle  has  one  right  or  obtuse 
angle,  the  other  two  angles  are  acute. 

Use  Th.  III. 

13.  In  an  isosceles  triangle  the  base  angles  are  acute. 

14.  The  sum  of   any  two  angles  of   any  triangle  is  less  than  a 
straight  angle. 

15.  If  from  a  point  a  perpendicular  and  other  lines  are  drawn  to  a 
given  line,  then  the  angle  formed  by  any  of  the  lines  on  its  side 
which  is  away  from  the  perpendicular,  is  obtuse,  and  the  greatest 
angles  in  the  figure  are  the  ones  formed  by  the  lines  farthest  from 
the  perpendicular  on  its  two  sides. 

16.  If  from  the  ends  of  a  side  of  a  triangle  lines  are  drawn  to  a 
point  within,  the  angle  formed  is  greater  than  the  angle  opposite  that 
side. 

100.  Theorem     IV.     //  two  sides  of  a  triangle  are 
equal,  the  angles  opposite  those  sides  are  equal. 

An  auxiliary  line  is  necessary,  but  the  analysis  will 
show  what  line  is  necessary.  What  method  of  proving 
angles  equal  uses  the  given,  that  is,  uses  equal  lines? 

17.  An  equilateral  triangle  is  equiangular. 


TRIANGLE   THEOREMS  57 

18.  The  bisectors  of  the  base  angles  of  an  isosceles  triangle  are 
equal. 

19.  The  medians  to  the  legs  of  an  isosceles  triangle  are  equal. 

20.  Two  points  on  the  base  of  an  isosceles  triangle  equally  dis- 
tant from  the  ends  of  the  base  are  equally  distant  from  the  vertex. 

21.  If  equal  distances  are  laid  off  on  the  sides  of  an  equilateral 
triangle  taken  in  order,  the  points  obtained  are  the  vertices  of  an 
equilateral  triangle. 

22.  Assuming  that  the  bisectors  of  the  base  angles  of  an  isosceles 
triangle  meet,  prove  that  they  form  an  isosceles  triangle. 

101.  COR.  1 .   In  an  isosceles  triangle,  the  bisector  of  the 
vertex  angle,  the  median  to  the  base,  the  altitude  to  the 
base,  and  the  perpendicular  bisector  of  the  base,  are  all 
one  line. 

The  proof  of  Th.  IV  can  be  extended  to  prove  this. 
It  is  very  largely  a  matter  of  determination  of  the  lines. 

102.  COR.  2.    There  can  be  but  two  equal  lines  from  a 
given  point  to  a  given  line. 

Draw  three  lines  supposed  to  be  equal,  and  examine  the 
base  angles. 

103-  Theorem  V.  If  two  sides  of  a  triangle  are  une- 
qual, the  angles  opposite  those  sides  are  unequal,  the 
greater  angle  being  opposite  the  greater  side. 

The  classification  shows  that  it  must  be  done  by  the 
inequality  axiom  or  by  exterior  angle.  Try  to  form  a 
new  angle  equal  to  one  of  the  angles  or  to  part  of  it,  but 
exterior  to  the  other. 

23.  In  the  quadrilateral  A  BCD,  if  AB  is  greater  than  EC,  EC 
than   CD,  and  CD  than  DA,  then  angle  D  is  greater  than  angle  B. 
Is  angle  C  always  greater  than  angle  A  ? 

24.  ABCD  is  a  quadrilateral  of  which  DA  is  the   longest  side, 
BC  the  shortest.     Prove  angle  B  greater  than  D,  and  C  than  A. 


58  RECTILINEAR  FIGURES 

104.  Theorem  VI.  (a)  If  two  angles  of  a  triangle  are 
equal,  the  opposite  sides  are  equal.  (&)  //  two  angles  of  a 
triangle  are  unequal,  the  opposite  sides  are  unequal,  the 
greater  side  being  opposite  the  greater  angle. 

This  is  evidently  the  converse  of  Th.  IV  and  Th.  V, 
and  since  they  cover  all  possibilities,  the  law  of  converse 
can  be  used.  The  following  form  is  recommended  for 
all  theorems  proved  by  the  law  of  converse. 


Given.     AABC;  /.A  =  Z#,  ZJ.  >ZJ3,  or  /.A</.B. 
To  prove.     BC  =  CA,    BC  >  CA,    or    BC  <  CA,    respec- 
tively. 

Proof.    I.   If  BC ;=  CA,  then  Z  A  =  Z B  (opp.  =  sides). 
If  BO  CA,  then  Z  A  >  Z  B  (opp.  =£  sides). 
If  BC<CA,  then  Z  A<Z..B  (opp.  Asides). 
II.   These  statements  cover  all  possibilities,  and 
no  two  of  the  conclusions  can  be  true  at  once. 
III.    .•.  If  Z  A  =  Z  J5,  then  BC=  CA. 
If  Z  A  >Z  B,  then  BC>  CA. 
If  Z  A  <  Z  B,  then  BC<  CA   (law  of  con- 
verse). 

25.  An  equiangular  triangle  is  equilateral. 

26.  In  a  right-angled  triangle,  the  hypotenuse  is  the  longest  side. 

27.  In  an  obtuse-angled  triangle,  the   side  opposite   the  obtuse 
angle  is  the  longest  side. 


TRIANGLE  THEOREMS  59 

28.  Assuming  that  the  bisectors  of  the  angles  of  a  triangle  which 
is   not  isosceles  meet,  prove  fche  bisector  of  the  smaller  angle  the 
longer  line. 

29.  Angle  A  of  triangle  ABC  is  bisected  by  a  line  meeting  BC 
at  P.     Prove  that  AB  is  longer  than  PB,  CA  than  PC. 

30.  If  a  perpendicular  and  two  other  lines  on  the  same  side  of 
the  perpendicular  are  drawn  from  a  point  to  a  given  line,  the  line 
which  is  farther  from  the  perpendicular  is  the  longer. 

105.  Theorem  VII.     If  two  triangles  have  three  sides 
of  one  equal  respectively  to  the  three  sides  of  the  other, 
the  triangles  are  congruent. 

Will  superposition  work  ?  If  not,  what  other  methods 
of  proving  triangles  congruent  are  known  ?  Don't  neglect 
the  given.  Read  Appendix,  §  344  (2). 

31.  In  an  equilateral  quadrilateral,  (1)  the  diagonals  bisect  the 
angles  and  are  the  perpendicular  bisectors  of  each  other;    (2)  the 
opposite  angles  are  equal. 

32.  Two  triangles  are  congruent  if  they  have  two  sides  and  the 
median  to  one  of  those  sides  respectively  equal. 

33.  If  one  diagonal  of  a  quadrilateral  divides  it  into  two.  isosceles 
triangles,  the  other  diagonal  bisects  two  of  the  angles  and  is  the 
perpendicular  bisector  of  the  first  diagonal ;  also,  one  pair  of  opposite 
angles  are  equal. 

106.  Theorem  VIII.     If  two  right  triangles  have  the 
hypotenuse  and  a  side  of  the  one  respectively  equal  to  the 
hypotenuse  and  a  side  of  the  other,  the  triangles  are  con- 
gruent. 

NOTE.  Investigate  also  the  congruence  of  triangles  having  two 
sides  and  an  angle  not  included  equal,  when  the  angle  is  not  right. 

34.  If  two  isosceles  triangles  have  the  equal  sides  of  one  the  same 
length  as  the  equal  sides  of  the  other,  and  the  base  of  one  double  the 
altitude  of  the  other,  the  triangles  are  equivalent. 

35.  If  a  quadrilateral  has  a  pair  of  opposite  sides  equal,  and  a  pair 
of  opposite  angles  right  angles,  one  diagonal  .divides  it  into  congruent 
triangles. 


60  RECTILINEAR  FIGURES 

107.  Theorem  IX.     The  sum  of  any  two  sides  of  a  tri- 
angle is  greater  than  the  third  side. 

Draw  the  sum  of  the  two  sides  in  the  simplest  position 
possible ;  then  classify. 

108.  COR.  I.     The  difference  of  any  two  sides  of  a  tri- 
angle is  less  than  the  third  side. 

If  AB  +  BC  >  CA,  why  is  CA  —  BC  <  AB  ? 

109.  COR.  II.     A  straight  line  between  two  points  is 
less  than  any  broken  line  between  those  points.     (See 
Appendix,  §  346.) 

36.   If  two  sides  of  a  triangle  are  6  ft.  and  10  ft.  long,  what  can  be 
told  about  the  length  of  the  third  side  V 

*  37.   If  two  sides  of  a  triangle  are  a  and  b,  what  can  be  told  of  the 
length  of  the  third  side  ? 

38.  The  sum  of  the  sides  of  a  triangle  is  greater  than  two  thirds 
the  sum  of  the  medians,  but  is  less  than  twice  that  sum. 

39.  The  sum  of  the  sides  of  a  quadrilateral  is  greater  than  the 
sum  of  the  diagonals,  but  is  less  than  twice  their  sum. 

40.  If  from  the  ends  of  a  side  of  a  triangle  lines  are  drawn  to  a 
point  within,  the  sum  of  the  lines  so  drawn  is  less  than  the  sum  of 
the  other  two  sides  of  the  triangle. 

41.  The  sum  of  the  lines  from  the  vertices  of  a  triangle  to  any 
point  is  greater  than  half  the  perimeter  of  the  triangle. 

42.  The  difference  of  the  diagonals  of  a  quadrilateral  is  less  than 
the  sum  of  either  pair  of  opposite  sides. 

110.  Incorrect    Proofs   depending  on  Location.  —  Many 
apparently  correct   proofs  are  entirely  incorrect,  or   are 
incomplete,  because  some  point  or  line  appears  to  be  in  a 
certain  position,  and  is  therefore  assumed  to  be  in  that  posi- 
tion in  the  proof,  when  as  a  matter  of  fact  it  is  never  in 
that  position  at  all,  or  is  in  that  position  for  certain  cases 
only. 


TRIANGLE   THEOREMS  61 

If  the  assumed  location  is  never  right,  the  proof  is  en- 
tirely incorrect ;  if  the  location  is  right  for  some  cases, 
but  not  for  all,  the  proof  applies  only  to  those  cases  for 
which  the  location  is  correct,  —  that  is,  it  is  not  a  correct 
general  proof.  On  this  account  great  care  should  be 
taken  that  no  point  or  line  is  assumed  to  be  in  a  certain 
position  unless  it  has  been  proved  to  lie  in  that  position 
always. 


EXAMPLE:  INCOMPLETE  PROOF  OF  THEOREM  IX 

Drop  a  perpendicular  from  c  to  AB  at  F. 

Then  ,  '         BC>  FB  (hyp.  rt.  A). 

CA  >  ^LF(hyp.  rt.  A). 

And  adding,  BC  +  CA  >  AF+  FB  (or  AB). 

The  fault  here  is  in  assuming  that  AF+FB  =  AB, 
which  is  true  only  if  F  falls  on  AB.  It  is  not  necessary 
that  F  shall  fall  on  AB,  for  the  perpendicular  can  just  as 
readily  meet  the  extension  of  AB. 

This  proof  can  be  completed  so  as  to  be  correct  for  all 
cases  .by  adding  a  proof  that  applies  to  the  case  where  the 
point  F  is  in  the  extension  of  AB. 

111.  Theorem  X.  If  two  triangles  have  two  sides  of  the 
one  equal  to  two  sides  of  the  other,  but  the  included  angles 
unequal,  then  the  third  sides  are  unequal,  the  greater  side 
being  opposite  the  greater  angle. 


62 


RECTILINEAR  FIGURES 


ANALYSIS 

Class.     Lines  unequal. 

Preceding  Methods.     Ineq.  ax. ;  opp.  uneq.  A ;  2  s.  >  3  d. 

Elimination.  The  inequality  axiom  has  been  used  to  ob- 
tain the  two  following  ways,  so  it  is  more  likely  that  it  will 
be  used  indirectly  through  them  than  that  it  alone  will 
give  the  proof.  Neither  of  the  other  methods  will  apply 
if  the  triangles  are  entirely  separate,  so  they  must  be  put 
together,  either  by  being  placed  side  by  side,  or  by  super- 
position. The  given  will  have  to  determine  which  method 
will  be  best.  The  fact  that  one  angle  is  greater  than  the 
other  can  be  used  by  cutting  off  the  smaller  on  the  greater, 
or  by  taking  enough  from  the  greater  so  that  when  it  is 
added  to  the  smaller,  the  angles  will  be  equal.  The  secpnd 
way  is  nothing  more  than  bisecting  the  sum  of  the  two 
angles.  It  is  possible  to  get  a  proof  by  following  any  of 
the  methods  mentioned,  but  the  following  way  is  perhaps 
the  easiest. 

Given.      A  ABC,  DEF-,   BC=EF,  CA  =  FD,Z.C>-Z.F. 

To  prove.     AB  >  DE. 


F    A 


K 


Application.  Let  the  triangles  be  placed  with  FD  coin- 
ciding with  its  equal  CA,  B  and  E  lying  on  opposite  sides 
of  the  common  line.  Draw  a  line  bisecting  the  angle 
BCE;  it  will  meet  the  opposite  side  AB  of  A  ABC,  for 


TRIANGLE   THEOREMS  63 

Z  C>  Z  jr,  and  so  this  line  will  lie  within  Z  c.  Call 
the  point  where  it  meets  AB,  K. 

Now  the  condition  has  changed  from  two  sides  equal 
and  the  included  angles  unequal,  to  two  sides  (#C  and  CK, 
and  EC  and  CK)  and  the  included  angles  (Z  BCK  and 
Z  ECK)  equal.  This  suggests  congruent  triangles,  so  the 
line  EK  is  drawn,  forming  AECE^ABCK.  Then  EK  = 
KB,  and  the  proof  by  two  sides  greater  than  the  third 
follows. 

NOTE.  This  proof  holds  even  when  EA  and  AB  happen  to  fall 
in  one  straight  line. 

WARNING.  "  opp.  >  Z,"  and  "opp.  >  s."  are  often  used  as  author- 
ity. It  must  be  kept  in  mind  that  this  applies  only  in  the  same  triangle, 
or  in  triangles  having  two  sides  equal.  Lines  and  angles  must  not  be 
judged  unequal  on  account  of  the  opposite  parts  in  any  other  cases. 

43.  In  triangle  ABC,  if  CA  is  greater  than  AB,  and  P  and  Q  are 
taken  on  AB  and  CA,  respectively,  so  that  PB  equals  QC,  prove  BQ 
less  than  CP. 

44.  If  a  quadrilateral  has  a  pair  of  opposite  sides  equal,  but  the 
angles  formed  by  those  sides  with  one  of  the  other  sides  unequal, 
then  the  diagonals  are  unequal. 

45.  If  no  median  of  a  triangle  is  perpendicular  to  a  side,  then  the 
triangle  has  no  equal  sides. 

46.  If  an  equilateral  pentagon  is  not  equiangular,  the  longest  diag- 
onal is  that  which  joins  the  vertices  on  either  side  of  the  greatest 
angle. 

112.  Theorem  XI.  //  two  triangles  have  two  sides  of 
the  one  equal  to  two  sides  of  the  other,  but  the  third  sides 
unequal,  then  the  angles  opposite  those  sides  are  unequal, 
the  greater  angle  being  opposite  the  greater  side. 

What  relation  has  this  to  Th.  X?  Is  Th.  X  all  that  is 
required  to  prove  this  theorem  ? 

47.  In  the  figure  of  exercise  44,  the  angles  formed  by  the  equal 
sides  with  the  fourth  side  are  also  unequal,  the  greater  angle  being 
opposite  the  greater  of  the  two  given  angles. 


64  RECTILINEAR  FIGURES 

48.  If  lines  are  drawn  from  the  ends  of  the  base  of  a  triangle  so  as 
to  cut  off  equal  distances  (from  the  base)  on  the  opposite  sides,  the 
triangle  is  isosceles  when  those  lines  are  equal;  when  those  lines  are 
unequal,  the  longer  line  is  drawn  to  the  shorter  side. 

113.  Theorem  XII.  •  Of  all  lines  drawn  to  a  given  line 
from  a  given  external  point 

(a)  the  perpendicular  is  the  shortest.  (See  Appendix, 
§346.) 

(5)  those  making  equal  angles  with  the  perpendicular, 
or  cutting  off  equal  distances  from  its  foot,  are  equal. 

(c)  those  making  unequal  angles  with  the  perpendicu- 
lar, or  cutting  off  unequal  distances  from  Us  foot,  are  un- 
equal, the  one  making  the  greater  angle,  or  cutting  off  the 
greater  distance,  being  the  greater. 

114.  COR.  1.    (a)  Equal  obliques  from  a  point  to  aline 
make  equal  angles  with  the  perpendicular  and  cut  off 
equal  distances  from  its  foot. 

(6)  Unequal  obliques  from  a  point  to  a  line  make  un- 
equal angles  with  the  perpendicular,  and  cut  off  unequal 
distances  from  its  foot,  the  longer  oblique  making  the 
greater  angle,  and  cutting  off  the  greater  distance. 


SECTION  IX.  PARALLELS  AND  PARALLELOGRAMS 

115.  Angles  formed  by  Two  Lines  and  a  Transversal. 
A  line  cutting  other  lines  is  called  a  transversal  of  those 
lines. 

If  two  lines  are  cut  by  a  transversal  at  two  points, 
eight  angles  are  formed  at  those  points,  and  certain  sets 
of  those  angles  have  names  / 

as  follows  :  / 

The  two  sets  of  angles     

in  the  figure  —  A  1,  3,  5,  7 
and  Zs  2,  4,  6,  8  — are  called 
transverse  sets  of  angles, 
and  any  two  angles  in  the  5/6 


71 1* 
v 

The  angles  Z  1,  Z  2,  Z  7,  Z  8  are  called  exterior  angles, 
and  the  angles  Z  3,  Z  4,  Z  5,  Z  6  are  called  interior  angles. 

The  pairs  of  angles  Z 1  and  Z  7  ;  Z  2  and  Z  8  ;  Z  3  and 
Z  5  ;  Z  4  and  Z  6  are  called  alternate  pairs  of  angles. 

The  pairs  Z  1  and  Z  5  ;  Z  2  and  Z  6  ;  Z  3  and  Z  7  ;  Z  4 
and  Z  8  are  called  corresponding  or  exterior  interior  angles. 
It  is  evident  that  any  pair  of  alternate  or  corresponding 
angles  are  also  in  the  same  transverse  set. 

116.  Theorem  XIII.  If  a  transversal  of  two  lines 
makes  one  pair  of  transverse  angles,  which  are  not  ver- 
tical, equal, 

(1)  any  two  transverse  angles  are  equal; 

(2)  any  two  angles  not  transverse  are  supplemental. 
SMITH'S  SYL.  PL.  OEOM.  —  5         65 


66  RECTILINEAR  FIGURES 

117.  COR.   1.    If  two  non-adjacent  angles,  which  are 
not  transverse,  are  supplemental, 

(1)  any  two  transverse  angles  are  equal ; 

(2)  any  two  angles  not  transverse  are  supplemental. 

NOTE.  These  propositions  are  introductory  to  the  subject  of 
parallels,  and  should  be  used  wherever  possible  to  shorten  the  fol- 
lowing proofs.  They  show  that  the  angles  formed  by  two  lines  and 
a  transversal  separate  into  two  groups  when  there  is  any  question  of 
equality  of  the  angles,  and  that  the  angles  of  different  groups  are 
supplemental  when  angles  of  the  same  group  are  equal. 

118.  Parallels.     Two  straight  lines  in  the  same  plane 
which   never    meet,   however    far    produced,    are    called 
parallel  lines. 

119.  Parallel  Axiom.     Through  a  given  point  there  can 
be  but  one  parallel  to  a  given  line. 

This  is  sometimes  stated  :  Two  intersecting  lines  cannot 
both  be  parallel  to  the  same  line. 

*  120.  Lines  parallel  to  the  same  line  are  parallel  to 
each  other.  For,  if  two  lines  that  are  parallel  to  the 
same  line  should  meet,  there  would  be  more  than  one  par- 
allel through  a  point. 

121 .    Theorem  XIV.     If  a  transversal  of  two  lines  makes 

(1)  a  pair  of  transverse  angles,  which  are  not  vertical, 
equal,  or 

(2)  two  non-adjacent  angles,  which  are  not  transverse, 
supplemental, 

then  the  two  lines  are  parallel. 

Is  there  any  difference  between  the  two  cases  ?  If  the 
lines  meet,  examine  the  angles  ;  use  contraposite. 


PARALLELS  AND  PARALLELOGRAMS      67 

122.  COK.   1.     If  a  transversal  of  two  lines  makes 

(1)  a  pair  of  transverse  angles  unequal, 

(2)  two  angles  which  are  not  transverse  not  supple- 
mental, then  the  lines  are  not  parallel. 

This  is  the  obverse  of  §  121.  The  best  way  to  prove  the 
obverse  of  a  single  statement  is  to  use  both  "  givens,"  and 
show  that  the  two  conclusions  could  not  be  the  same. 
Applied  to  this  statement,  the  work  would  be  about  as 
follows  : 

If  a  pair  of  transverse  angles  were  equal,  the  lines 
would  be  parallel. 

If  a  second  line  through  the  same  point  on  the  trans- 
versal makes  an  angle  not  equal  to  the  other  of  the  pair, 
that  line  cannot  also  be  parallel,  by  the  parallel  axiom. 
This  is,  of  course,  in  very  condensed  form  ;  the  proof 
should  be  in  terms  of  the  letters  of  the  figure. 

Note  that  the  lines,  if  not  parallel,  meet  on  the  side  of 
the  smaller  of  the  two  alternate  interior  angles.  Why  ? 

123.  COR.  2.    Lines  perpendicular  to  the  same  line  are 
parallel. 

49.  The  bisector  of  the  exterior  vertex  angle  of  an  isosceles  tri- 
angle is  parallel  to  the  base. 

50.  If  A  C  and  BD  bisect  each  other,  prove  that  AB  is  parallel  to 
CD. 

51.  The  bisectors  of  two  consecutive  angles  of  any  polygon  inter- 
sect one  another. 

52.  If  all  the  angles  of  a  quadrilateral  are  right  angles,  the  figure 
has  two  pairs  of  parallel  sides. 

124.  Theorem  XV.     If  two  parallels  are  cut  by  a  trans' 
versal, 

(1)  any  two  transverse  angles  are  equal; 

(2)  any  two  angles  that  are  not  transverse  are  sup- 
plemental. 


68  RECTILINEAR  FIGURES 

What  relation  has  this  to  Th.  XIV?  to  Th.  XIV,  Cor. 
1  ?  After  proving  this  by  the  simplest  way,  see  Appendix, 
§341. 

125.  COR.   1.     A  line  perpendicular  to  one  of  two  par- 
allels is  perpendicular  to  the  other. 

It  is  necessary  first  to  prove  that  it  meets  the  other; 
in  other  words  to  prove  that  it  cannot  be  parallel  to  it. 
If  it  were  parallel  to  one  of  the  two,  while  given  perpen- 
dicular to  the  other,  what  impossible  figure  would  be 
formed  ? 

WARNING.  Never  assume  that  two  lines  meet  without  proving  why 
they  meet. 

126.  COK.    2.     Two  lines  that  are  perpendicular  to  two 
intersecting  lines  are  not  parallel. 

127.  COR.    3.     If  the  arms  of  one  angle  are  parallel  to 
the  arms  of  a  second  angle,  the  angles  are  equal,  or  supple- 
mental (according  to  their  relative  positions) . 

128.  COR.  4.     If  the  arms  of  one  angle  are  perpendicu- 
lar to  the  arms  of  a  second  angle,  the  angles  are  equal,  or 
supplemental  (according  to  their  relative  positions). 

Through  the  vertex  of  one  of  the  angles  draw  parallels 
to  the  arms  of  the  other  angle,  thus  using  angles  at  one 
vertex  (§  127). 

53.  The  bisectors  of  a  pair  of  alternate  angles  formed  by  parallels 
with  a  transversal  are  parallel. 

54-  A  line  through  the  vertex  of  an  isosceles  triangle  parallel  to 
the  base  bisects  the  exterior  angle  at  the  vertex. 

55.  If  the  bisector  of  an  exterior  angle  of  a  triangle  is  parallel  to 
a  side  of  the  triangle,  the  triangle  is  isosceles. 

56.  If  from  a  point  in  the  base  of  an  isosceles  triangle,  lines  are 
drawn  parallel  to  the  equal  sides,  those  lines  form,  with  the  equal 
sides,  a  quadrilateral  whose  perimeter  equals  twice  one  of  the  equal 
sides. 


PARALLELS  AND  PARALLELOGRAMS  69 

57.  A  line  parallel  to  the  base  of  an  isosceles  triangle  makes  equal 
angles  with  the  legs,  or  the  legs  produced. 

58.  A  line  parallel  to  the  base  of  an  isosceles  triangle  is  perpen- 
dicular to  the  bisector  of  the  vertex  angle. 

129.  Theorem  XVI.     In  any  triangle, 

(1)  any  exterior  angle  equals  the  sum  of  the  two  non- 
adjacent  interior  angles  ; 

( '2)  the  sum  of  all  the  interior  angles  equals  a  straight 
angle. 

What  is  known  about  the  exterior  angle?    Use  it. 

130.  COR.   1.      If  two  triangles  have  a  side  and  any 
two  angles  of  the  one  equal  to  a  side  and  the  respective 
angles  of  the  other,  the  triangles  are  congruent.     Use  the 
theorem. 

This  is  the  last  of  the  propositions  on  the  congruence  of 
triangles  by  equal  parts.  It  should  be  noticed  that  three 
parts  are  always  necessary,  and  that  any  three  correspond- 
ing parts  will  prove  the  triangles  congruent  except 

(1)  three  angles.     (See  §§  286,  289.) 

(2)  two  sides  and  an  acute  angle  not  included  (called 
the  Ambiguous  Case).     (See  §106.) 

131.  COR.  2.     If  a  triangle  has  one  right  angle,  the 
other  two  angles  are  complements ;  in  an  equilateral  tri- 
angle, each  angle  is  one  third  of  a  straight  angle. 

59.  Using  (1)  of  §  129,  prove  (2)  by  drawing  a  line  from  the  vertex 
to  any  point  in  the  base. 

60.  The  bisectors  of  two  interior  angles  of  a  triangle  meet  (ex- 
amine the  angles  made  with  the  included  side),  and  form  an  angle 
equal  to  a  right  angle  plus  one  half  the  third  angle. 

61.  The  bisectors  of  two  exterior  angles  of  a  triangle  meet,  and 
form  an  angle  equal  to  one  half  the  sum  of  the  interior  angles  supple- 
mental to  the  ones  bisected. 


70  RECTILINEAR  FIGURES 

68.  If  an  angle  of  a  triangle  is  bisected,  and  a  line  is  drawn  per- 
pendicular to  that  bisector,  that  line  makes  an  angle  with  either  arm 
of  the  bisected  angle  equal  to  half  the  sum  of  the  other  angles  of  the 
triangle,  and  makes  an  angle  with  the*  third  side  equal  to  half  the 
difference  of  those  angles. 

63.  Any  two  altitudes  of  a  triangle  make  equal  angles  each  with 
the  side  to  which  the  other  is  drawn. 

64-  Find  the  sum  of  the  angles  of  a  quadrilateral  by  drawing  a 
diagonal. 

132.  Theorem  XVII.     The  sum  of  the  interior  angles  of 
a  polygon  of  n  sides  is  (n  —  2}  straight  angles. 

65.  What  is  the  sum  of  the  interior  angles  of  a  pentagon  ?  a  hexa- 
gon? a  decagon?  an  octagon?  a  29-sided  figure? 

66.  How  many  sides  has  the  polygon  the  sum  of  whose  interior 
angles  is  12  st.  angles?  3  st.  angles?  17  st.  angles? 

67.  How  large  is  one   angle   of  a   regular  pentagon?  hexagon? 
decagon  ?  octagon  ?  34-sided  figure  ? 

133.  Theorem  XVIII.      The  sum  of  the  exterior  angles  of 
a  polygon  is  two  straight  angles. 

68.  How  many  sides  has  the  polygon  the  sum  of  whose  interior 
angles  equals  the  sum  of  the  exterior  angles? 

69.  How  many  sides   has  a  polygon  if  the  sum  of  the  interior 
angles  is  seventeen  times  the  sum  of  the  exterior  angles? 

70.  How  many  sides  has  the  polygon  one  third  the  sum  of  whose 
interior  angles  is  ten  times  the  sum  of  its  exterior  angles? 

71.  How  many  sides  has  the  polygon  each  of  whose  exterior  angles 
is  one  tenth  of  a  perigon  ? 

72.  How  many  sides  has  the  polygon  each  of  whose  interior  angles 
is  six  sevenths  of  a  straight  angle  ? 

134.  Degrees,  Minutes,  and  Seconds.     For  convenience 
in  numerical  calculations,  a  perigon  is  diyided  into  360 
equal  parts,  called  degrees;   each  degree  is  divided  into 


PARALLELS   AND  PARALLELOGRAMS  71 

60  equal  parts  called  minutes,  and  each  minute  into  60 
equal  parts  called  seconds. 

73.  How  many  degrees  are  there  in  an  interior  angle  of  a  regular 
polygon  of  15  sides  ?  of  n-sides? 

74.  How  many  degrees  are  there  in  an  exterior  angle  of  a  regular 
polygon  of  20  sides?  of  n-sides? 

135.  Quadrilaterals  having  Two  or  More  Parallel  Sides. 

A  quadrilateral  having  two  pairs  of  parallel  sides  is  called 
a  parallelogram ;  one  which  has  one  pair  of  parallel  sides 
is  called  a  trapezoid ;  one  which  has  no  parallel  sides  is 
called  a  trapezium. 

The  parallel  sides  of  a  trapezoid  are  called  its  bases,  the 
other  two  sides  its  legs. 

Either  pair  of  parallel  sides  may  be  called  the  bases  of  a 
parallelogram,  but  the  side  on  which  it  appears  to  stand  is 
usually  called  its  base. 

A  parallelogram  having  all  its  angles  right  angles  is 
called  a  rectangle;  one  having  all  its  sides  equal  is  called 
a  rhombus ;  a  rectangular  rhombus  is  called  a  square.  A 
parallelogram  that  is  neither  a  rectangle  nor  a  rhombus  is 
called  a  rhomboid. 

136.  Theorem   XIX.     Any  two  consecutive  angles  of  a 
parallelogram  are  supplemental,  and  any  two  opposite 
angles  are  equal. 

137.  COR.  1.     A  parallelogram  having  a  right  angle  is 
a  rectangle. 

138.  Theorem  XX.     In  any  parallelogram,  • 

(1)  either  diagonal  divides  it  into  congruent  triangles; 

(2)  the  opposite  sides  are  equal. 

139.  COR.  1.     The  diagonals  of  a  parallelogram  bisect 
each  other. 


72  RECTILINEAR   FIGURES 

75.  The  diagonals  of  a  parallelogram  are  equal  if  the  figure  is  a 
rectangle. 

76.  A   parallelogram    having  two   consecutive    sides  equal   is    a 
rhombus. 

77.  Two  trapezoids  are  congruent  if  their  sides  are  respectively 
equal. 

140.  Theorem  XXI.     A  quadrilateral  is  a  parallelo- 
gram if 

(1)  two  opposite  sides  are  equal  and  parallel;  or 

(2)  the  opposite  sides  are  equal. 

What  is  the  definition  of  a  parallelogram  ?  What  ways 
of  proving  this  do  you  know?  Keep  in  mind  that  the 
given  must  be  used. 

141.  Theorem  XXII.     A  quadrilateral  is  a  parallelo- 
gram if  the  opposite  angles  are  equal. 

142.  Theorem  XXIII.     A  quadrilateral  is  a  parallelo- 
gram if  the  diagonals  bisect  each  other. 

78.  Js  a  quadrilateral  a  parallelogram  if  one  diagonal  divides  it 
into  congruent  triangles?  if  each  diagonal  divides  it  into  congruent 
triangles? 

79.  If  each  side  of  a  square  is  extended  from  each  end  a  length 
half  as  long  as  the  diagonal,  the  figure  formed  by  joining  the  ends  of 
the  sects  obtained  is  a  regular  octagon. 

80.  On  diagonal  BD  of  parallelogram  A  BCD,  points  K  and  L  are 
taken  so  that  BK  —  LD.     Prove  that  AKCL  is  a  parallelogram. 

81.  Perpendiculars  from   opposite  vertices  of   a  parallelogram  to 
the  diagonal  joining  the  other  vertices  are  equal. 

82.  Lines  through  the  vertices  of  a  triangle  parallel  to  the  oppo- 
site sides  form  a  triangle  having  sides  twice  the  length  of  the  sides  of 
the  given  triangle. 

83.  Lines  drawn  through  the  vertices  of  a  quadrilateral  parallel  to 
the  diagonals  form  a  parallelogram  twice  as  large  as  the  quadrilateral. 


PARALLELS  AND  PARALLELOGRAMS       73 

84.  Any  quadrilateral  is  equal  to  a  triangle  having  as  sides  the 
diagonals  of  the  quadrilateral,  and  the  included  angle  equal  to  the 
angle  between  the  diagonals. 

85.  The  sum  (or  the  difference)  of  the  perpendiculars  from  one 
pair  of  opposite  vertices  of  a  parallelogram  to  any  line  equals  the  sum 
(or  the  difference)  of  the  perpendiculars  from  the  other  vertices  to 
the  same  line. 

86.  The  sum  of  the  perpendiculars  dropped  from  a  point  in  the 
base  of  an  isosceles  triangle  to  the  legs  is  constant. 

87.  The  difference  of  the  perpendiculars  from  any  point  in  the 
extension  of  the  base  of  an  isosceles  triangle  to  the  legs  is  constant. 

143.  Theorem  XXIV.    If  two  or  m,ore  sects  cut  off  on  one 
transversal  by  parallels  are  equal,  the  corresponding  sects 
cut  off  on  any  other  transversal  by  the  same  parallels  are 
also  equal;  including  as  a  special  case, 

A  line  through  the  midpoint  of  a  side  of  a  triangle, 
parallel  to  a  second  side,  bisects  the  third  side. 

144.  COR.  1.    A  line  joining  the  midpoints  of  two  sides 
of  a  triangle  is  parallel  to  the  third  side,  and  equals  one 
half  that  side. 

145.  COR.  2.    If  through  the  vertices  of  a  triangle, 
lines  are  drawn  parallel  to  the  opposite  sides,  a  new  tri- 
angle is  formed,  such  that  the  sides  of  the  given  triangle 
join  the  midpoints  of  the  sides  of  the  new  triangle. 

88.  The  lines  from  two  opposite  vertices  of  a  parallelogram  to  the 
midpoints  of  two  parallel  sides  trisect  the  diagonal  joining  the  other 
two  vertices. 

89.  The  lines  joining  the  midpoints  of  the  opposite  sides  of  a  quad- 
rilateral bisect  each  other. 

When  do  lines  bisect  each  other?     What  way  is  there  of  using 
midpoints  to  obtain  this  ? 

90.  The  lines  of  89  intersect  at  the  midpoint  of  the  line  which 
joins  the  midpoints  of  the  diagonals. 


74  RECTILINEAR  FIGURES 

91.  A  line  through  the  midpoint  of  one  of  the  legs  of  a  trapezoid, 
parallel  to  the  bases,  bisects  the  other  leg. 

92.  The  line  joining  the  midpoints  of  the  legs  of 'a  trapezoid  is 
parallel  to  the  bases. 

93.  What  line  passes  through  the  midpoints  of  all  the  lines  drawn 
from  a  point  to  a  line? 

94.  If,  in  an  isosceles  triangle,  any  number  of  parallels  to  the  base 
are  drawn,  the  perpendicular  bisector  of  the  base  goes  through  the 
intersections  of  the  diagonals  of  each  trapezoid  formed. 

NOTE.  The  trapezoid,  while  not  used  in  the  theorems,  has  been 
taken  up  quite  frequently  in  exercises.  Many  of  the  triangle  theorems 
have  corresponding  facts  for  trapezoids,  and  on  this  account,  a  good 
way  to  study  the  trapezoid  is  to  word  the  triangle  theorems  so  that 
they  apply  to  trapezoids,  and  investigate  their  truth.  Probably  the 
most  valuable  auxiliary  line  for  trapezoids  is  the  line  from  the  vertex 
between  the  shorter  base  and  a  leg,  parallel  to  the  other  leg.  This 
line  divides  the  trapezoid  into  a  triangle  and  a  parallelogram,  and  so 
allows  the  theorems  for  those  figures  to  be  applied. 


SECTION   X.     LOCI   OF    POINTS   AND   CONCURRENCE 

146.  Locus.     That  place,  or  places,  in  which  all  points 
which  fulfill  a  certain  condition  and  no  others  lie,  is  called 
the  locus  of  points  which  fulfill  that  condition. 

If  all  pupils  of  a  certain  class  are  in  a  certain  room 
(that  is,  no  pupil  is  absent),  and  all  in  the  room  belong 
to  that  class  (that  is,  there  are  no  visitors  present),  then 
that  room  contains  all  pupils  of  that  class  and  no  others. 
Similarly,  if  a  line  contains  every  point  of  a  certain  kind 
(where  "  kind  "  means  in  respect  to  some  condition),  and 
every  point  on  it  is  of  that  kind,  then  it  is  the  locus  of 
such  points.  Evidently  a  locus  is  nothing  but  the  place 
which  may  be  said  to  belong  to  a  certain  kind  of  point. 

To  prove  a  locus,  it  is  necessary  to  prove  two  things  : 

(1)  all  points  of  the  kind  lie  in  the  place  ; 

(2)  all  points  in  the  place  are  of  the  same  kind. 
These  statements  are  converses  of  each  other,  and  their 

contraposites  will  also  be  true. 

(3)  all  points  not  in  this  place,  are  not  of  this  kind ; 

(4)  all  points  not  of  this  kind  do  not  lie  in  this  place. 
It  is  sometimes  easier  to  prove  (3)  or  (4)  instead  of  its 

contraposite,  the  most  common  way  being 'to  prove  (3) 
instead  of  (1). 

147.  Attack  of  a  Locus  Theorem.     The  best  way  to  dis- 
cover a  locus  theorem  is  to  take  a  point  of  the  required 
kind  and  study  it  for  characteristics  which  will  show  that 
it  must  lie  in  some  certain  place.     After  the   place   is 

75 


76  RECTILINEAR  FIGURES 

found,  it  is  also  necessary  to  prove  that  every  point  in  the 
place  is  of  the  same  kind. 

148.  Theorem  XXV.     Find  the  locus  of  points  equidis- 
tant frojn  two  given  points. 

Take  any  point  equidistant  from  two  given  points,  join 
the  three  points,  and  what  kind  of  a  triangle  will  be 
formed?  What  will  pass  through  the  vertex?  Then 
what  is  the  place  which  appears  to  be  the  locus  ?  Now 
prove  that  all  points  on  the  line  found  are  equidistant 
from  the  given  points. 

149.  Con.  1.     A  line  containing  two  points  that  are 
equidistant  from  the  ends  of  a  sect  is  the  perpendicular 
bisector  of  that  sect. 

150.  COR.  2.     The  midpoint  of  the  hypotenuse  of  a 
right  triangle  is  equidistant  frojn  tlxe  three  vertices. 

95.  Find  the  locus  of  points  equidistant  from  three  points.     (Use 
the  points  two  at  a  time.) 

151.  Concurrence.     If   two   or   more    lines    meet   in   a 
point,  the  lines  are  said  to  be  concurrent  at  that  point. 

152.  Theorem  XXVI.     Tl:e  perpendicular  bisectors  of 
the  three  sides  of  a  triangle  are  concurrent. 

Show  that  two  meet,  then  that  the  third  must  pass 
through  the  same  point. 

This  point  of  concurrence  is  equidistant  from  the  three 
vertices,  and  is  "called  the  circumcenter  of  the  triangle. 
(See  §  165.) 

96.  If  the  circumcenters  of  the  triangles  formed  by  a  diagonal  of 
a  parallelogram  are  joined  to  the  ends  of  the  diagonal,  a  parallelo- 
gram is  formed. 


•LOCI   OF  POINTS  AND   CONCURRENCE  77 

97.  The  circumcenters  of  the  four  triangles  into  which  a  quadri- 
lateral is  divided  by  its  diagonals  are  the  vertices  of  a  parallelogram. 

98.  In  triangle  ABC,  if  the  midpoints  of  BC  and  CA  are  joined, 
the  circumcenters  of  the  original  triangle  ($)  and  of  the  new  triangle 
(/$*')  and  C  are  in  one  straight  line,  and  #'  is  the  midpoint  of  the 
line. 

153.  Theorem  XXVII.     Find  the  locus  of  points  equi- 
distant from  two  intersecting  lines. 

NOTE.  Locus  does  not  always  mean  equidistance,  although  it 
involves  equidistance  in  the  first  few  cases. 

99.   Find  the  locus  of  points  equidistant  from  two  parallel  lines. 

100.  Find  the  locus  of  points  at  a  fixed  distance  from  a  given  line. 

101.  If  two  points  between  the  arms  of  an  angle  are  equidistant 
from  those  arms,  the  line  through  the  points  is  the  bisector  of  the 
angle. 

102.  Find  the  locus  of  points  equidistant  from  three   lines   that 
intersect  in  no  point;  in  one  point;  in  two  points ;  in  three  points. 

103.  Find  the  locus  of  points  equidistant  from  two  intersecting 
lines,  and  at  the  same  time  equidistant  from  two  points. 

104.  Find  the  locus  of  points  equidistant  from  two  given  lines 
and  at  a  fixed  distance  from  a  given  line. 

154.  Theorem   XXVIII.     The  bisectors  of  the  interior 
angles  of  a  triangle  are  concurrent. 

The  point  of  concurrence  is  called  the  incenter,  and 
it  is  equidistant  from  the  three  sides  of  the  triangle. 
(See  §  166.) 

155.  COR.  1.     The  bisectors  of  any  two  exterior  angles 
of  a  triangle,  and  the   third  interior  angle,    are  con- 
current. 

There  are  three  such  points  of  concurrence,  all  equi- 
distant from  the  sides  of  the  triangle ;  they  are  called 
excenters.  (See  §  166.) 

Of  what  are  the  incenter  and  the  excenters  the  locus  ? 


78  RECTILINEAR  FIGURES 

105.  If  the  incenter  of  an  equilateral,  triangle  is  joined  to  the 
vertices,  the  triangles  formed  are  congruent. 

106.  If  the  incenters  of  the  triangles  formed  in  105  are  joined  to 
each  other,  the  incenter  of  the  original  triangle  is  the  circumcenter 
of  the  new  triangle. 

107.  If   the   incenters   of  the  triangles  into   which    a    diagonal 
divides  a  parallelogram  are  joined  to  the  ends  of  the  diagonal,  a 
parallelogram  is  formed. 

108.  The  incenters  and  the  circumcenters  of  the  triangles  into 
which   a  diagonal   divides    a   parallelogram   are  the   vertices   of   a 
p  aralle  logr  am . 

156.  Theorem  XXIX.     The  altitudes  of  a  triangle  are 
concurrent. 

Use  §  145.     The  point  is  called  the  orthocenter. 

109.  In  triangle  ABC,  if   YX  joins  the  midpoints  of  BC  and  CA, 
and  O  and  0'  are  the  orthocenters  of  triangles  ABC  and  XCY,  then 
O,  0',  and  C  are  collinear,  and  CO'  =  00'. 

110.  In  the  figure  o£  109,  if  S  and  S'  are  the  circumcenters,  /  and 
/'  the  incenters,  then  O'S1  is  parallel  to  OS  and  equals  one  half  OS, 
O'l'  is  parallel  to  OI  and  equals  one  half  OI. 

157.  Theorem  XXX.     The  medians  of  a  triangle  are 
concurrent  in  a  trisection  point  of  each. 

Show  that  two  meet.  Draw  the  Hue  from  the  third 
vertex  through  their  intersection,  and  show  that  it  bisects 
the  third  side,  and  is  therefore  the  third  median.  This 
line  may  be  extended  its  own  length  past  the  intersection 
point,  and  shown  to  be  one  diagonal  of  a  parallelogram. 

The  point  of  concurrence  is  called  the  centroid. 

111.  The  sum  of  the  three  medians  of  a  triangle  is  greater  than 
three  fourths  the  sum  of  the  sides. 

112.  In  triangle  AB C,  if  YX  joins  the  midpoints  of  BC  and  CA, 
and  if  T  and  T  are  the  centroids  of  triangles  ABC  and  XCY,  T,  T, 
and  C'  are  collinear,  and  CT1  =  TT'. 


SUMMARY  OF   PROPOSITIONS  79 

113.  In  the  figure  of  112,  if  £',  /,  0',  and  S,  /,  0,  are  the  circum- 
center,  incenter,  and  orthocenter,  respectively,  of  triangles  JTGTand 
ABC,  then  T  S'  is  parallel  to  TS  and  equals  one  half  of  it;  TT  is 
parallel  to  TI and  equals  one  half  of  it;  TO'  is  parallel  to  TO  and 
equals  one  half  of  it. 

158.      SUMMARY  OF  THEOREMS  AND  COROLLARIES,  BOOK  I 

(Numbers  in  parentheses  refer  to  black- faced  section  numbers.) 

I.  CONGRUENCE  OF  TRIANGLES.     Triangles  are  congruent  when 
they  have  three  parts  (of  which  at  least  one  is  a  side)  of  one  equal  to 
the  corresponding  parts  of  the  other,  except  when  the  parts  are  two 
sides  and  an  acute  angle  not  included  between  those  sides. 

Cases.  2  s.  incl.  Z  (92)  ;  2  A  incl.  s.  (94)  ;  3  s.  (105)  ;  rt.  Z,  hyp., 
leg  (106)  ;  2  A  any  s.  (130)  ;  diag.  par.  (138). 

II.  LINES  ARE  DETERMINED,  OR"  COINCIDE.    One  _L  from  a  point 
(97)  ;   lines  in  an  isosceles  A  (101)  ;  one  I!  through  a  point  (119). 
(See  also  89.) 

III.  LINES  EQUAL.     Congruence;  opp.  =  A in  a  A  (104);  obliques 
(113, 114)  ;  opp.  s.  n  (138) ;  diag.  O  bisect  (139)  ;  II  s  cutting  off  equal 
sects  on  transversals  (143) ;  lines  through  midpt.  of  s.  of  a  A  (144, 
145) ;  midpt.  hyp.  rt.  A  equidist.  (150). 

IV.  LINES  UNEQUAL.     Opp.  =£  A  in  a  A  (104)  ;  but  two  =  lines 
from  a  pt.  (102)  ;  sum  of  two  s.  of  a  A  >  3d,  diff.  less  (107, 108, 109)  ; 
opp.  =£  A  in  A  having  two  =  s.  (Ill) ;  obliques  (113,  114). 

V.  ANGLES  EQUAL,  AND  ANGLE  EQUATIONS.      Opp.  =  s.  (100)  ; 
obliques  (114)  ;  Us  (116, 117, 124)  ;  II  arms  (127)  ;  ±  arms  (128)  ;  opp. 
A  a  (136,  137).    (See  also  89.) 

Ext.  Z  of  a  A  =  two  int.  A,  sum  A  of  a  A  =  st.  Z  (129)  ;  acute  A 
of  a  rt.  A  =  rt.  Z  (131) ;  int.  A  of  a  polygon  =  (n  -  2)  st.  A  (132)  ; 
ext.  A  =  2  st.  A  (133). 

VI.  ANGLES  UNEQUAL.     Ext.  Z  of  a  A  (95)  ;  in  rt.  or  obtuse  angled 
A  other  A  acute  (99)  ;  opp.  =£  s.  in  a  A  (103)  ;  opp.  =£.  s.  in  two  A 
having  two  =  s.  (112)  ;  obliques  (114). 

VII.  ANGLES  SUPPLEMENTAL.     Us  (116,  117,  124)  ;  II  arms  (127)  ; 
A.  arms  (128)  ;  con.  sec.  A  of  a  O  (136). 


80  RECTILINEAR   FIGURES 

VIII.  LINES  PARALLEL.   A  =  (121);  A  sup.  (121);  JL  same  line 
(123) ;    II  same  line  (120)  ;   opp.  s.  O.     (See  XI  of  this  set) ;   line 
through  midpts.  of  the  s.  of  a  A  (144.) 

IX.  LINES  NOT  PARALLEL.     But  one  ||  through  a  pt.  (119)  ;  A  =£ 
(122)  ;  A  not  sup.  (122)  ;  ±  intersecting  lines  (126). 

X.  LINES  PERPENDICULAR.     Line  _L  one  of  two  Us  _L  other  (125); 
contains  two  points  equidist.  (149).     (See  also  89.) 

XL  A  QUADRILATERAL  A  PARALLELOGRAM.  Opp.  s.  ||  (135); 
opp.  s.  =  (140) ;  two  opp.  s.  =  and  ||  (140) ;  opp.  A  =  (141)  ;  diag. 
bisect  (142). 

XII.  Loci.     Points  equidist.  from  two  points  (148);  equidist.  from 
two  intersecting  lines  (153).     (See  exercises  99,  100.) 

XIII.  CONCURRENCE.    _L  bisectors  of  sides  (152)  ;  bisectors  int.  A 
(154);   bisectors  2  ext.  A  and  int.  (155);  altitudes  (156)  ;  medians 
(157). 

These  classifications  should  be  carefully  studied,  and  made  use  of 
in  solving  exercises  depending  on  the  respective  classes.  The  pupil 
should  not  depend  on  the  list  here,  but  should  know  the  important  ways 
of  doing  theorems  of  each  class. 

159.  ORAL  AND  REVIEW  QUESTIONS 

What  are  the  two  most  common  ways  of  proving  lines  equal  ?  the 
three  most  common  of  proving  angles  equal?  the  ways  to  prove  lines 
parallel  ?  not  parallel  ?  a  figure  a  parallelogram  ?  Name  the  points  of 
concurrence  of  the  sets  of  lines  in  a  triangle.  What  is  known  about 
the  circumcenter  ?  the  incenter?  the  excenters?  the  centroid?  If 
you  wish  to  prove  that  two  lines  bisect  each  other,  how  ought  you 
to  start?  If  it  is  necessary  to  prove  two  lines  equal,  and  two  angles 
are  given  equal,  how  are  they  likely  to  be  used  ?  How  can  correspond- 
ing parts  of  congruent  triangles  be  distinguishe'd  without  superim- 
posing? What  would  be  the  sum  of  the  interior  angles  of  a  polygon 
of  K  sides  ?  of  79  sides  ?  the  exterior  angles  of  each  ?  If  the  interior 
angles  of  a  polygon  have  a  sum  of  29  straight  angles,  how  many  sides 
has  the  polygon?  a  sum  of  (2  K  +  6)  right  angles?  If  the  interior 
angles  of  a  polygon  are  eleven  times  the  exterior  angles,  how  many 


ORAL   AND   REVIEW   QUESTIONS  81 

sides  has  the  polygon  ?  If  each  exterior  angle  is  one  twenty-second 
of  a  straight  angle,  how  many  sides  has  the  polygon?  If  each  interior 
angle  is  nineteen  twentieths  of  a  straight  angle,  how  many  sides  has 
the  polygon  ?  If  two  sides  of  a  triangle  are  17  and  28,  how  long  is 
the  third  side  ?  How  many  points  need  to  be  known  to  determine  a 
perpendicular  bisector?  What  ways  are  known  to  determine  a  line? 
In  how  many  ways  should  the  same  line  be  determined?  What  class 
of  theorems  would  be  used  to  prove  that  two  lines  meet?  Outline 
in  as  brief  a  manner  as  possible  the  methods  used  in  proving  the  fol- 
lowing :  base  angles  of  an  isosceles  triangle  are  equal ;  sum  of  interior 
angles  of  a  polygon  ;  quadrilateral  having  two  pairs  of  equal  sides  is 
a  parallelogram ;  lines  perpendicular  to  the  same  line  are  parallel ; 
but  one  perpendicular  from  a  point  to  a  line ;  exterior  angle  of  a  tri- 
angle equals  the  sum  of  the  non-adjacent  interior  angles;  opposite 
angles  of  a  parallelogram  are  equal ;  opposite  sides  of  a  parallelogram 
are  equal;  angles  having  parallel  arms  are  equal  or  supplemental; 
perpendicular  bisectors  of  the  sides  of  a  triangle  are  concurrent. 

NOTE.  Be  ready  to  explain  why  each  construction  used  in  these 
proofs  is  the  best  one  to  use. 

If  the  sides  of  a  triangle  are  a,  b,  c,  give  air  the  length  relations 
possible  between  the  sides.  How  many  equal  lines  are  there  from  a 
point  to  a  line  ?  Which  statement  (converse,  obverse,  or  contraposite) 
always  follows  from  a  true  statement?  which  sometimes  follows? 
when  ?  State  three  places  where  converse  is  used  to  prove  theorems 
or  corollaries.  Try  to  explain  why  these  propositions  are  not  in 
reverse  order.  Give  some  examples  of  contraposite  reasoning  used  in 
proving  theorems.  What  is  the  locus  of  points  equidistant  from  three 
points  not  in  the  same  line?  Can  the  locus  of  points  equidistant  from 
four  or  more  points  be  found?  Why?  From  four  or  more  lines? 
Why?  What  are  the  two  most  important  things  to  keep  in  mind  in 
trying  to  prove  a  theorem  ?  which  should  be  thought  of  first,  in  gen- 
eral ?  Why  is  it  necessary  to  be  careful  of  the  location  of  points  and 
lines? 

GENERAL  EXERCISES 

114'  If  two  lines  drawn  from  the  ends  of  the  base  of  an  isosceles 
triangle  to  the  opposite  sides  cut  off  equal  distances  from  the  vertex, 
those  lines  are  equal. 

115.  If  one  angle  of  a  triangle  is  twice  a  second  angle,  the  bisector 
of  the  larger  angle  forms  a  triangle  equiangular  to  the  given  triangle. 
SMITH'S  SYL.  PL.  GEOM.  —  6 


82  RECTILINEAR   FIGURES 

116.  If  one  of  the  opposite  sides  of  a  quadrilateral  is  the  longest 
side,  the  other  the  shortest  side,  examine  the  size  of  the  opposite 
angles. 

117.  The  sum  of  the  lines  from  the  vertices  of  a  triangle  to  any 
point  is  greater  than  one  half  the  perimeter  of  the  triangle. 

118.  The  sum  of  the  lines  from  the  vertices  of  a  quadrilateral  to 
any  point  is  greater  than  the  sum  of  the  diagonals. 

119.  The  difference  of  the  diagonals  of  a  quadrilateral  is  less  than 
half  the  sum  of  its  sides. 

120.  If  two  points  K  and  L  are  on  the  same  side  of  the  line  AB, 
to  what  point  P  in  AB  must  lines  be  drawn  from  K  and  L,  so  that 
KP  +  LP  shall  be  the  shortest  possible  length?     (Use  symmetry.) 

121.  If  a  pair  of  base  angles  of  a  trapezoid  are  equal,  the  legs  are 
equal,  and  conversely. 

122.  If  the  legs  of  a  trapezoid  are  unequal,  the  base  angles  are 
unequal,  and  conversely. 

123.  The  diagonals  of  an  isosceles  trapezoid  are  equal. 

124--  The  lines  joining  the  midpoints  of  the  consecutive  sides  of  a 
quadrilateral  form  a  parallelogram.  Examine  the  special  cases,  such 
as  parallelogram,  rhombus,  rectangle,  square. 

125.  Find   out  everything    possible    about    the    diagonals    of    a 
rhombus. 

126.  The  perpendicular  bisector  of  a  side  of  a  triangle  intersects 
the  greater  of  the  other  two  sides,  if  the  triangle  is  not  isosceles. 

127.  In  a  right  triangle,  the  altitude  to  the  hypotenuse  divides  the 
triangle  into  two  triangles  whose  angles  equal  the   angles  of  the 
original  triangle. 

128.  The  line  through  the  midpoints  of  the  bases  of  an  isosceles 
trapezoid  passes  through  the  intersection  of  the  diagonals,  and  the 
intersection  of  the  legs. 

129.  Two  lines  through  a  point  equidistant  from  two  parallels  cut 
off  on  the  parallels,  if  they  meet  them,  equal  sects. 

130.  The  midpoint  of  any  transversal  of  two  parallels  is  equidis- 
tant from  them. 

131.  Lines  drawn  from  any  point  in  the  bisector  of   an   angle 
parallel  to  its  arms  form  a  rhombus  with  those  arms. 


GENERAL   EXERCISES  83 

132.  A  line  through  the  incenter  of  a  triangle  parallel  to  a  side 
equals  the  sum  of  the  sects  it  cuts  off  on  the  other  two  sides. 

138.  If  the  three  sides  of  one  triangle  are  parallel  respectively  to 
the  three  sides  of  another  triangle,  the  angles  of  the  first  triangle  are 
respectively  equal  to  the  angles  of  the  second  triangle. 

134.  If  the  sides  of  one  triangle  are  respectively  perpendicular  to 
the  sides  of  a  second  triangle,  the  angles  of  the  first  triangle  are 
respectively  equal  to  the  angles  of  the  second  triangle. 

135.  The  angle  between  the  bisectors  of  two  exterior  angles  of  a 
triangle  equals  the  complement  of  half  the  third  interior  angle. 

136.  The  angle  between  the  bisectors  of  one  interior  and  one  ex- 
terior angle  of  a  triangle  equals  half  the  interior  angle  at  the  third 
vertex. 

137.  If  the  vertex  of  an  isosceles  triangle  is  joined  to  the  trisec- 
tion  points  of  the  base,  the  middle  of  the  three  angles  formed  is  the 
largest. 

138.  If  one  acute  angle  of  a  triangle  is  double  another,  the  tri- 
angle can  be  divided  into  two  isosceles  triangles  by  a  line  through 
the  third  vertex. 

139.  Two  exterior  angles  of  a  triangle  less  the  third  interior  angle 
equals  a  straight  angle. 

140.  The  angle  between  the  bisectors  of  two  angles  of  a  triangle 
equals  the  third  angle  plus  half  the  sum  of  the  bisected  angles. 

141.  Two  equilateral  triangles  are  congruent  if  an  altitude  of  one 
equals  an  altitude  of  the  other. 

142.  An  angle  of  a  triangle  is  acute,  right,  or  obtuse,  according  as 
the  median  from  it  is  greater  than,  equal  to,  or  less  than  one  half  the 
opposite  side. 

143.  The  angle  between  the  bisectors  of  the  base  angles   of  an 
isosceles  triangle  equals  the  exterior  angle  at  the  base. 

144.  If  equal  distances  are  measured  off  on  the  sides  of  a  square 
taken  in  order,  the  points  obtained  will  be  the  vertices  of  a  second 
square. 

145.  A  transversal  of  the  legs  of -an  isosceles  triangle,  one  pro- 
duced, which  is  perpendicular  to  the  base,  forms  an  isosceles  triangle 
with  the  legs. 


84  RECTILINEAR  FIGURES 

146.  Divide  the  right  angle  of  a  right  triangle  into  parts  equal  to 
the  acute  angles,  and  so  prove  that  the  midpoint  of  the  hypotenuse 
is  equidistant  from  the  three  vertices. 

147.  If  one  of  the  equal  sides  of  an  isosceles  triangle  is  extended 
through  the  vertex  its  own  length,  the  line  joining  the  end  of   the 
extension  to  the  other  end  of  the  base  is  perpendicular  to  the  base. 

148.  If  the   angles  of   a   quadrilateral   are   bisected,  the  angles 
formed  by  the  bisectors  of   any  pair  of   adjacent   angles  is  supple- 
mental to  the  angle  formed  by  the  bisectors  of  the  other  angles. 

149.  A  trapezoid  whose  diagonals  make  equal  angles  with  a  base 
is  isosceles. 

150.  Prove  the  theorem  for  the  sum  of  the  angles  of  a  polygon 
by  joining  one  vertex  to  all  the  others.     How  many  diagonals  can  be 
drawn  in  any  n-sided  polygon  ?     How  many  triangles  will  be  formed 
in  an  n-sided  polygon  ? 

151.  How  many  sides  has  a  polygon  the  sum  of  whose  interior 
angles  is  four  times  the  sum  of  the  interior  angles  of  a  quadrilateral? 

152.  How  many  sides  has  a  polygon  the  sum  of  whose  interior 
angles  is  equal  to  the  sum  of  the  interior  angles  of  a  hexagon  plus 
the  sum  of  the  exterior  angles  of  an  octagon  ? 

153.  The  "triangles  formed   by  producing  the  sides  of   an  equi- 
angular hexagon  through  both  vertices  of  each  side  are  equilateral. 

154-   The  sum  of  the  angles  formed  by  producing  the  sides  of  a 
hexagon  both  ways  until  they  meet  equals  a  perigon. 

155.  If  the  sides  of   any  polygon  of   w-sides  are   produced    from 
both  ends  until  they  meet,  the  sum  of  the  angles  so  formed  is  n  —  4 
straight  angles. 

156.  If  each  consecutive  pair  of  angles  of  a  quadrilateral  are  sup- 
plemental, is  the  figure  a  parallelogram  ? 

157.  A  parallelogram  which  has  supplemental  opposite  angles  is 
a  rectangle. 

158.  Is  a  quadrilateral  a  parallelogram  if   one  pair   of   opposite 
sides  are  equal,  and  the  other  pair  are  parallel  ? 

159.  If  each  side  of  a  square  is  produced  in  order,  the  extensions 
of  the  opposite  sides  being  equal,  the  points  obtained  are  the  vertices 
of  a  parallelogram. 


GENERAL   EXERCISES  85 

160.  The  sum  of  the  perpendiculars  from  any  point  to  the  three 
sides  of  an  equilateral  triangle  is  constant.     (In  certain  cases  some 
of  the  lines  become  negative,  and  the  difference  would  be  taken.) 

161.  If  X  and  Y  cut  off  equal  sects  from  A  and  C  on  the  sides 
AB  and  CD  of  the  parallelogram  A  BCD,  AXCY  is  a  parallelogram. 

162.  Parallelograms   having  two  consecutive  sides  and  an  angle 
of  one  equal  to  the  corresponding  parts  of  the  other  are  congruent. 

163.  Rectangles  having  two  consecutive  sides  of  the  one  equal  to 
the  corresponding  parts  of  the  other  are  congruent. 

164-   ID  anv  parallelogram,  a  line  through  the  midpoint  of  a  side 
parallel  to  a  consecutive  side  divides  it  into  two  congruent  parts. 

165.  The  median  to  any  side  of  a  triangle  is  less  than  one  half  the 
sum  of  the  other  two  sides. 

166.  The  midpoints  of  two  opposite  sides  of  any  quadrilateral  and 
of  its  diagonals  are  the  vertices  of  a  parallelogram.      What  if   the 
given  figure  is  a  parallelogram  ? 

167.  A  line  drawn  from  A  of  triangle  ABC,  through  the  mid- 
point of  the  median  to  CA,  to  K  on  BC,  cuts  off  one  third  of  BC. 

168.  If  three  parallels  are  drawn  from  the  ends  and  the  midpoint 
of  a  sect  to  meet  a  given  line,  the  parallel  from  the  midpoint  is  half 
the  sum  of  the  parallels  from  the  ends. 

169.  If  equal  sects  are  cut  off  on  one  leg  of  an  isosceles  triangle 
and  the  other  leg  extended  through  the  base,  the  line  joining  the 

•points  obtained  is  bisected  by  the  base. 

170.  A  line  from  a  point  on  a  side  of  an  equilateral  triangle  to  a 
point  twice  as  far  from  the  common  vertex  on  another  side  is  per- 
pendicular to  one  of  the  sides  of  tlve  triangle. 

171.  A  line  from  the  vertex  of  the  right  angle  of  a  right  triangle 
to  the  middle  of  the  hypotenuse  divides  the  triangle  into  two  isosceles 
triangles. 

172.  The  bisectors  of  the  two  interior  adjacent  angles  formed  by 
a  transversal  with  one  of  two  parallels  cut  off  equal  sects  from  the 
transversal  on  the  other  parallel. 

173.  If  two  consecutive    sides  of   a   parallelogram  are   extended, 
not  from  the  common  vertex,  their  own  lengths,  the  line  joining  their 
extremities  passes  through  the  fourth  vertex. 


86  RECTILINEAR  FIGURES 

174.  One  angle  of  a  triangle  is  double  another;  the  included  side 
is  constant,  but  the  angles  change.  Find  the  locus  of  the  intersection 
of  the  bisector  of  the  larger  angle  with  the  opposite  side. 

175-  If  the  circumcenters  of  the  triangles  at  the  vertices,  formed 
by  joining  the  midpoints  of  the  sides  of  a  triangle,  are  joined  to  the 
midpoints  of  the  two  sides  which  help  form  each  triangle,  an  equi- 
lateral hexagon  (not  always  convex)  is  formed. 

176.  In  triangle  ABC,  if  YX  joins  the  midpoints  of  the  sides  BC 
and  CA,  the  incenters  7  and  /',  of  triangles  ABC  and  XCY,  and  C 
are  collinear,  and  CI'  =  II'. 

177.  In  the  figure  of  176,  if  S  and  Sf  are  the  circumcenters,  ST 
is  parallel  to  SI,  and  equals  one  half  SI. 

NOTE.  Compare  Exercises  109,  110,  112,  113,  176,  177,  and  note 
that  the  facts  proved  of  each  corresponding  pair  of  points  are  the 
same.  Upon  what  one  fact  about  the  figure  do  all  these  proofs 
depend  ? 


BOOK   II.     CIECLES 

SECTION   I.     DEFINITIONS 

160.  Circles.  The  limited  portion  of  a  plane  bounded 
by  a  closed  line,  every  point  of  which  is  equidistant  from 
a  point  within  called  the  center,  is  a  circle.  The  closed 
line  that  bounds  a  circle  is  called  its  circumference;  the 
term  "circumference"  is  often  used  for  the  length  of  the 
circumference,  the  context  usually  showing  which  meaning 
is  to  be  used.  The  amount  of  surface  in  a  circle  is  called 
the  area  of  the  circle.  In  the  advanced  subjects  of  Mathe- 
matics, the  word  "  circle  "  is  used  for  the  circumference, 
and  some  books  on  Geometry  are  now  using  "  circle  "  in 
that  sense.  The  meaning  is  usually  clear,  and  the  pupil 
should  not  be  confused  by  either  use.  Some  of  the  exer- 
cises will  be  worded  in  each  way,  although  in  the  text  the 
definitions  will  be  adhered  to. 

*Jl  circumference  is  a  curve. 

For  if  any  part  were  straight,  it  could  not  have  all  its 
points  equidistant  from  the  center  (§  102). 

*A  circle  has  but  one  center. 

For  if  it  had  two,  they  would  both  be  midpoints  of  the 
sect  through  them  with  its  ends  on  the  circumference. 

A  circle  is  named  by  the  letter  at  its  center,  unless  that 
would  cause  ambiguity ;  as,  Oo  would  mean  the  circle 
having  its  center  at  O. 

A  part  of  a  circumference  is  called  an  arc. 

87 


88  CIRCLES 

161.  Radii,  Diameters,  Chords.    A  line  from  the  center  of 
a  circle  to  any  point  on  the  circumference  is  called  a  radius. 

A  line  joining  two  points  of 
a  circumference  is  called  a 
chord;  a  chord  through  the 
center  is  called  a  diameter. 

OB  is  a  radius,  AB  is  a  diam- 
eter, CD  is  a  chord. 

The  parts  of  the  circumfer- 
ence, BKA,  ACDB,  are  arcs,  and 
are  named,  in  the  direction 
contrary  to  that  taken  by  the 
hands  of  a  clock,  BA,  AB. 

*  Radii  of  the  same  circle  are  equal  (§  160). 

*  Diameters  of  the  same  circle  are  equal. 

162.  A  point  is  within  a  circle,  on  the  circumference,  or 
outside  the  circle,  according  as  its  distance  from  the  center 
is  less  than,  equal  to,  or  greater  than,  a  radius. 

163.  Equality   of   Circles.     If   two  circles  have   equal 
radii,  they  are  congruent,  for  if  superimposed  they  coin- 
cide.    If  two  circles  are  equivalent,  they  are  congruent, 
for  if,  when  superimposed  with  the  centers  coinciding,  any 
point  of  the  circumference  of  one  fell  within  the  other, 
all  points  of  that   circumference  would   fall  within    the 
other,  and  the  circle  would  be  a  part  of  the  other,  and 
therefore  smaller. 

On  this  account,  equal  will  be  used  for  circles  that  are 
equivalent,  or  congruent,  or  have  equal  radii,  for  there  is 
no  distinction  between  these  cases. 

From  the  foregoing  it  is  evident  that  but  one  circum- 
ference can  be  drawn  with  a  given  center  and  a  given 
radius, 


DEFINITIONS  89 

A  circle  is  sometimes  supposed  to  be  rotated  about  its 
center,  so  that  any  particular  point  of  the  circle  takes  a 
different  position.  The  circle  as  a  whole  would  evidently 
be  in  the  same  position  as  before  rotation,  but  chords,  arcs, 
radii,  etc.,  would  have  changed  position.  This  can  be 
used  to  superimpose  one  figure  in  the  circle  on  the  place 
occupied  by  another  figure.  (In  this  case,  one  figure 
would  be  supposed  to  rotate,  while  the  other  would 
remain  in  the  same  position.) 

*164.  Through  three  points  not  in  the  same  straight 
line,  one,  and  but  one  circumference  can  be  drawn 
(§  152).  Points  in  the  same  straight  line  are  called 
collinear. 

165.  Inscribed  and  Circumscribed  Figures.     A  circum- 
ference that  passes  through  the  vertices  of  a  polygon  is 
said  to  be  circumscribed  about  the  polygon,  and  the  poly- 
gon is  said  to  be  inscribed  in  the  circle. 

A  circumference  that  lies  within  a  polygon  and  touches 
each  of  its  sides  in  one  point  only  is  said  to  be  inscribed 
in  the  polygon,  and  the  polygon  is  said  to  be  circumscribed 
about  the  circle. 

§  164  might  have  been  stated  One,  and  but  one,  cir- 
cumference can  be  circumscribed  about  a  given  triangle. 

166.  One,  and  but  one,  circumference  can:  be  inscribed 
in  a  given  triangle. 

See  §  154.  Note  (by  §  155)  that  three  circumferences 
can  be  drawn  outside  a  triangle,  touching  the  sides  each  in 
one  point.  Such  circumferences  are  said  to  be  escribed  to 
the  triangle. 

167.  Intersection  of  Lines  and  Circumferences.     A  line 
that  meets  a  circumference  at  one    point   only  is   called 
a  tangent  to  the  circle ;  one  that  meets  the  circumference 


90  CIRCLES 

at  two  points  is  called  a  secant.  The  point  where  a  tan- 
gent meets  a  circumference  is  called  the  point  of  tangency, 
or  point  of  contact. 

(1)  *A  line  perpendicular  to  a  radius  at  its  intersection 
with  the  circumference  is  a  tangent.     For  all  other  points 
on  the  line  are  at  a  distance  from  the  center  greater  than 
the  radius. 

(2)  *  A  line  which  meets  a  circumference  at  a  point, 
and'  is  not  perpendicular  to  the  radius  to  that  point,  is  a 
secant. 

For  a  perpendicular  can  be  drawn  to  it,  and  a  second 
oblique  can  be  found  equal  to  the  radius,  thus  showing 
a  second  point  of  intersection  with  the  circle  (§  113). 

From  the  closed-figure  intersection  axiom,  and  the  pre- 
ceding statement,  it  follows  that  (3)  a  line  at  less  than  a 
radius  distance  from  the  center  of  a  circle  —  that  is,  a  line 
through  a  point  within  a  circle  —  is  a  secant. 

(4)  *A   tangent  to  a  circle  is  perpendicular  to  the 
radius  drawn  to  the  point  of  contact.    Follows  from  (2). 

(5)  *The  perpendicular  to  a  tangent  at  the  point  of  con- 
tact passes  through  the  center  of  the  circle. 

(6)  *A  line  at  more  than  a  radius  distance  from  the 
center  of  a  circle  does  not  meet  the  circumference. 

168.  Center   Lines.     The   line    through  the  centers  of 
two  circles  is  called  their  center  line  ;  that  part  of  it  be- 
tween the  centers  is  called  their  center  sect. 

169.  Length  of  the  center  sect  dependent  on  the  position 
of  the  two  circles  in  regard  to  each  other. 

(1)  *//  two  circumferences  do  not  meet,  their  center 
sect  is  greater  than  the  sum,  or  less  than  the  difference,  of 
the  radii,  according  as  the  center  of  one  is  outside,  or  in- 
side, the  other. 


DEFINITIONS  91 

(2)  *If  two  circumferences  meet  on  their  center  line, 
the  center  sect  equals  the  sum,  or  the  difference,  of  the 
radii,  according  as  the  center  of  one  circle  is  outside,  or 
inside,  tJw  other. 

(3)  *  If  two  circumferences  meet  at  a  point  not  on  tlwir 
center  line,  the  center  sect  is  less  than  the  sum,  and  greater 
than  the  difference,  of  the  radii. 

If  the  figures  are  drawn,  the  reasons  for  the  statements 
will  be  found  to  depend  upon  the  inequality  axiom,  and 
the  length  relations  between  the  sum  or  difference  of  two 
sides  of  a  triangle  and  the  third  side. 

170.    Intersection   of  Two  Circumferences.     (Converses 

of  §  169.) 

*If  the  center  sect  of  two  circles 

(1)  is  greater  than  the  sum  of  the  radii,  the  circumfer- 
ences do  not  meet,  and  lie  outside  each  other. 

(2)  is  less  than  the  difference  of  the  radii,  the  circum- 
ferences do  not  meet,  and  one  circle  is  within  the  otJwr. 

(3)  equals  tlie  sum  of  the  radii,  the  circumferences 
meet  on  the  center  line,  and  are  outside  of  each  other. 

(4)  equals  the  difference  of  the  radii,  the  circumfer- 
ences meet  on  the  center  line,  and  one  circle  is  inside  the 
other. 

(5)  is  less  than  the  sum  and  greater  than  the  difference 
of  the  radii,  the  circumferences  meet  at  a  point  not  on  the 
center  line. 

These  statements  are  used  in  all  cases  where  it  is  neces- 
sary to  show  whether  or  not  two  circumferences  meet. 

tfoxE.  There  are  five  values  for  the  center  sect:  less  than  the 
difference  of  the  radii,  equal  to  that  difference,  between  the  difference 
and  the  sum,  equal  to  the  sum,  and  greater  than  the  sum.  The  pupil 
should  understand  the  results  from  each  of  these  lengths. 


92  CIRCLES 

171.  *Two  circumferences  that  meet  at  a  point  not  on 
their  center  line,  intersect  twice. 

If  the  circum- 
ferences of  Os  O 
and  of  meet  at  P, 
a  second  triangle 
OO'Q  could  be  con-  0 
structed  congruent 
to  OO'P,  by  drawing 
angles  on  the  other  Q 

side  of  O0r  equal  respectively  to  Z  o'OP  and  Z  oofP. 
Why  does  this  show  that  Q  is  a  second  point  of  inter- 
section ? 

NOTE.  To  show  that  two  circumferences  intersect  twice  it  is 
necessary  to  show. 

(1)  the  center  sect  less  than  the  sum  of  the  radii. 

(2)  the  center  sect  greater  than  the  difference  of  the  radii. 

172.  Common  Chord.     If  two  circumferences  meet  twice, 
the  line  joining  the  points  of  intersection  of  the  circles  is 
called  their  common  chord. 

*If  two  circumferences  intersect  twice,  their  center  line 
is  the  perpendicular  bisector  of  their  common  chord. 
Use  equidistance. 

173.  Tangency  of  Circles.      Two    circumferences    that 
meet  at  one  point  only  are  said  to  be  tangent  to  each  other, 
or  to  touch  each  other.     The  point  where  they  meet  is 
called  the  point  of  tangency,  or  the  point  of  contact.     The 
circles  also  are  sometimes  spoken  of  as  tangent  to  each 
other. 

*If  two  circumferences  meet  at  a  point  on  their  center 
line,  they  are  tangent  to  each  other.  If  they  meet  at  a 


ORAL  AND  REVIEW  QUESTIONS  93 

second  point  on  the  center  line,  they  are  identical ;  if  at 
a  second  point  which  is  not  on  the  center  line,  the  center 
sect  would  have  two  different  lengths,  which  is  impos- 
sible [§  169  (2),  (3)]. 

*Two  circles  tangent  to  the  same  line  at  the  same  point 
are  tangent  to  each  other. 

174.  The  shortest  sect  from  a  given  point  to  a  circum- 
ference is  a  sect  of  the  line  from  that  point  through  the 
center.     Why  ?     Prove  it  when   the  point  is  inside  and 
when  it  is  outside. 

175.  ORAL   AND   REVIEW   QUESTIONS 

If  the  radii  of  two  circles  are  10  ft.  and  6  ft.  long,  tell  what  is 
known  about  the  position  of  the  circles  if  the  center  sect  is  (1)  12  ft. 
long,  (2)  16  ft.  long,  (3)  4  ft.  long,  (4)  2  ft.  long,  (5)  20  ft.  long. 
Why  is  a  circumference  a  curve?  Why  has  it  but  one  center?  How 
many  points  not  in  a  straight  line  determine  a  circumference?  Can 
a  circumference  always  be  drawn  through  four  points?  Can  it  always 
be  drawn  tangent  to  three  lines?  When  are  circles  tangent?  When 
is  a  line  tangent  to  a  circle?  What  is  known  about  the  center  line 
of  two  intersecting  circumferences  and  their  common  chord?  What 
relation  is  there  between  the  distance  of  a  line  from  the  center  of  a 
circle  and  the  intersection  of  the  line  and  circumference.  If  it  is 
necessary  to  draw  two  circles  which  are  tangent  externally  (i.e.  out- 
side of  each  other),  how  long  must  the  center  sect  be  made?  If  it 
is  necessary  to  have  their  circumferences  meet  twice? 


SECTION   II.     CONSTRUCTIONS 

176.  Postulates.     In  plane  geometry,   the  use  of  two 
instruments   for   construction   is   allowed,  or  postulated. 
These  instruments  are  the  straight  edge  (or  ruler,   except 
that    no    measurements    may   be   taken    with   it)    and    the 
compass. 

The  postulates  that  allow  the  use  of  these  instruments 
are: 

(1)  A  straight  line  may  be  drawn  from  any  one  point  to 
any  other  point. 

(2)  A  sect  may  be  produced  to  any  length  in  that  line. 

(3)  A  circumference  may  be  described  with  any  point  as 
a  center  and  any  sect  as  a  radius. 

177.  Constructions.     In  the  theorems  of  Book  I,  aux- 
iliary lines  have  been  added  to  the  given  figure,  but  only 
as  representations  of  existing  lines  about  which  it  was 
necessary  to  reason  in  order  to  establish  the  proof.     The 
question  of   the  accuracy  of   the  drawing  of   these  lines 
had   no   bearing   on   the   truth   of   the  theorem,  for  the 
reasoning  was   entirely  about  the  figures  that  the  lines 
were   taken   to   represent.     In  the  problems  that  are  to 
be  done,  the  proof   consists  in  showing   that   the   figure 
constructed   is  —  within   the   limits   of    accuracy   of    the 
instruments    used — a   correctly   drawn   figure  according 
to   the   requirements  of   the   proposition.     The  theorem 
proves  a  fact ;  the  construction  makes  a  required  figure, 
then  proves  the  correctness  of  the  method  of  construction. 

94 


CONSTRUCTIONS  95 

178.  Analysis    of   a    Construction.       To    discover    the 
method  of  drawing   a   required  construction,   first  draw 
the   completed  figure  as  accurately  as  possible,   without 
actually    constructing    it.       By    the    study    of    the   com- 
pleted figure,  attempt  to  find  what   lines  must  be  con- 
structed   in   order   to   make   the    figure    in    such    a   way 
that   it   can   be   shown    to   be    the   required   one.      This 
is  really  working  backwards   from  the  completed  figure 
in   an   attempt   to   find   upon    what   it   is   based.     Hav- 
ing  analyzed   the    figure,    draw    the    lines   found   neces- 
sary, and  so  build  up  a  figure  that  can  be  proved  to  be 
correct. 

Do  not  neglect  the  classification  method,  for  it  will 
make  the  analysis  method  unnecessary  in  many  cases. 
For  example,  a  perpendicular  can  be  obtained  by  finding 
two  points  equidistant  from  the  ends  of  the  sect  to  which 
the  line  is  to  be  perpendicular. 

179.  Construction  Uses  of  the  Circumference.     The  cir- 
cumference is  the  locus  of  points  at  a  certain  fixed  distance 
from  a  fixed  point.     The  definition  shows  this  fact,  and  it 
is  of  great  importance,  both  in  proofs,  and  in  construction 
work.     The  two  following  hints  will  show  common  uses 
of  the  circumference. 

To  draw  a  line  of  a  given  length  from  a  given  point, 
use  the  compass  to  draw  a  circumference  around  the  given 
point,  with  the  given  sect  as  a  radius.  This  is  called 
describing  a  circumference  with  a  certain  center  and 
radius. 

To  construct  a  point  equidistant  from  two  given  points, 
use  each  of  the  given  points  as  a  center,  and  use  the  same 
radius.  What  relation  must  hold  between  the  center  sect 
and  the  radii? 


96 


CIRCLES 


178.  Construct  an  equilateral  triangle  on  a  given  sect  as  a  side. 

179.  Construct  a  triangle  having  its  sides  equal  to  three  given 
sects.     Is  it  always  possible? 

180.  Construct  an  angle  equal  to  two  thirds  of  a  right  angle. 

181.  Construct  an  isosceles  triangle  on  a  given  base,  having  each 
leg  twice  the  length  of  the  base. 

182.  Construct  a  regular  hexagon  on  a  given  sect  as  a  side. 

183.  Construct  three  equal  circles,  each  tangent  to  the  other  two. 
184-    Construct  seven  equal  circles,  six  of  them  surrounding  the 

seventh  and  each  tangent  to  three  others. 

185.  Construct  two  sects,  given  their  sum  and  their  difference. 

186.  Construct  a  sect,  given  a  second  sect  and  the  sum  of  the  two 
sects. 

187.  Construct  a  sect,  given  a  second  sect  and  the  difference  of 
the  two  sects. 

180.    CONSTRUCTION  I.    To  draw  the  perpendicular  bi- 
sector of  a  given  sect. 

ANALYSIS 

Two  points  equidistant  from    the   ends  of   the   given 
sect  will  determine  the  perpendicular  bisector. 

PROOF 


•B 


CONSTRUCTIONS 


97 


Given. 
Required. 

Construction.    I. 


II, 


III. 


To  prove. 
Proof. 


II. 


AB. 

To  draw  the  JL  bisector  of  AB. 

With  center  A,  and  radius  AB  (or  any 
radius  >  \  AB)  describe  a  circum- 
ference. With  center  B  and  the 
same  radius  describe  a  second  cir- 
cumference (circum.  post.). 

These  circumferences  meet  at  two 
points.  (Center  sect  is  less  than 
the  sum,  but  greater  than  the  dif- 
ference of  the  radii.)  Call  the 
points  of  intersection  P  and  Q. 

Draw  PQ  (line  post.). 

PQ  _L  AB  at  its  midpoint. 
PA  =  PB,     QA  =  QB    (radii    of    equal 
circles). 

.  • .  P  and  Q  are  equidistant  from  the 
ends  of  AB,  and  PQ  _L  AB  at  its 
midpoint  M.  (A  line  through  two 
points  equidistant  from  the  ends 
of  a  sect  is  the  perpendicular  bi- 
sector of  that  sect.) 


188.  Find  by  construction  the  point  equidistant  from  three  given 
non-collinear  points. 

189.  Describe  a  circumference  having  its  center  in  a  given  circle, 
and  passing  through  two  given  points. 

190.  Draw  a  line  through  a  given  point,  so  that  it  will  be  equi- 
distant from  two  other  given  points. 

191.  Construct  an  isosceles  triangle,  given  the  base  and  the  length 
of  the  altitude. 

SMITH'S  SYL.  PL.  GEOM. — 7 


98  CIRCLES 

181.  CONST.   II.      To  draw  a  perpendicular  to  a  given 
line  through  a  given  point 

(1)  in  the  line  ; 

(2)  outside  the  line. 
Use  equidistance. 

192.  Construct  an  isosceles  triangle,  given  the  altitude  and  a  leg. 

193.  Construct  a  triangle,  given 

(1)  Two  sides  and  the  altitude  to  the  third  side. 

(2)  The  base,  one  of  the  sects  cut  off  on  the  base  by  the  altitude, 
another  side. 

(3)  The  base,  its  median,  and  its  altitude. 

194-   Construct  a  rectangle,  given  a  side  and  a  diagonal. 

195.  Given  an  angle  and  a  point,  draw  a  line  through  the  point, 
so  as  to  make  equal  angles  with  the  arms  of  the  given  angle. 

196.  Construct  the  point  in  a  given  line,  such  that  lines  from  two 
given  points  on  the  same  side  of  the  given  line  to  the  point  found 
may  make  equal  angles  with  the  given  line. 

182.  CONST.  III.      To  draw  a  perpendicular  to  a  given 
sect  at  the  end,  without  extending  the  sect. 

Let  the  sect  be  a  leg  of  the  required  right  triangle, 
and  use  the  fact  known  about  the  midpoint  of  the  hypote- 
nuse. 

183.  CONST.  IV.      To  bisect  a  given  angle. 

Draw  the  lines  so  that  the  two  parts  of  the  angle  may 
be  proved  to  be  equal ;  that  is,  use  angles  equal. 

197.  Construct  the  points  that  are  equidistant  from  three  lines 
that  are  neither  concurrent  nor  all  parallel. 

198.  Construct  an  angle  equal  to  one  half  a  right  angle  ;  to  one 
third  a  right  angle. 

NOTE.  It  is  impossible  to  trisect  any  given  angle  with  no  instru- 
ments except  the  compass  and  ruler.  Such  exercises  as  the  second 


CONSTRUCTIONS  99 

part  of  198  must  be  done  by  using  some  angle  that  is  known  to  be 
one  third  of  a  right  angle,  or  that  can  be  made  into  one  third  of  a 
right  angle  by  bisection. 

199.  Construct  a  point  in  one  side  of  a  given  triangle  equidistant 
from  the  other  sides. 

184.  CONST.  V.     At  a  given  point  in  a  given  line,  to  con- 
struct a  line  making  a  given  angle  with  the  given  line. 

What  is  the  simplest  way  to  prove  two  angles  in  differ- 
ent places  equal  ? 

200.  Construct  a  triangle,  given  two  sides  and  the  included  angle. 

201.  Construct  a  triangle,  given  two  angles  and  the  included  side. 

202.  Construct  a  triangle,  given  two  angles  and  any  side. 

203.  Construct  an  isosceles  triangle,  given  the  altitude  and  the 
vertex  angle. 

204.  Construct  an  isosceles  triangle,  given  the  altitude  and  a  base 
angle. 

205.  Construct  a  right  triangle,  given  the  perimeter  and  either  leg. 

185.  CONST.  VI.      Through  a  given  point  to  draw  a  par- 
allel to  a  given  line. 

206.  From  a  point  outside  a  given  line,  draw  a  line  making  a 
given  angle  with  that  line. 

207.  Construct  a  triangle,  given  an  altitude  and  two  angles. 

208.  Construct  a  triangle,  given  an  angle,  one  of  the  sides  includ- 
ing the  angle,  the  median  from  the  same  vertex.     (Use  the  midpoint 
of  the  given  side.) 

209.  Draw  a  sect  terminated  by  the  arms  of  a  given  angle,  and 
through  a  given  point  within  the  angle,  so  that  the  point  will  bisect 
the  sect. 

210.  Draw    a  line  from   a   given    point   outside   a  given  angle, 
through  the  arms  of  the  angle,  so  one  of  the  arms  will  bisect  the  sect 
from  the  point  to  the  other  arm. 


100  CIRCLES 

211.  Construct  a  rhombus,  given 
(a)  A  side  and  an  altitude. 

(6)  The  altitude  and  a  diagonal. 

(c)  The  sum  of  the  altitude  and  a  side,  an  angle. 

186.  CONST.  VII.     To  divide  a  given  sect  into  any  num- 
ber of  equal  parts.     (Use  a  system  of  parallels.) 

212.  Divide  a  given  sect  so  that  one  part  of  it  equals  twice  the 
other  part;  so  that  one  part  equals  three  fifths  of  the  other. 

218.  Divide  a~given  sect  so  that  the  second  of  three  parts  equals 
twice  the  first,  the  third  equals  three  times  the  second. 

214.  Divide  a  given  sect  into  four  parts,  so  that  the  second  equals 
twice  the  first,  the  third  equals  three  times  the  first,  the  fourth  equals 
four  times  the  first. 

187.  Sectors  and  Segments.     A  sector  of  a  circle  is  that 
surface  bounded  by  two  radii  and  the  arc  between  their 
ends.     Two    radii   evidently  divide  the    circle   into   two 
sectors.     The  order  of  lettering  (reverse  to  the  direction 
taken  by  the  hands  of   a  clock)  shows  which  sector  is 
meant. 

'A  segment  of  a  circle  is  that  surface  bounded  by  a  chord 
and  the  arc  between  its  ends.  A  chord  evidently  divides 
the  circle  into  two  segments.  Here  again,  the  order  of 
lettering  shows  which  one  is  meant. 

188.  Angles  in  a  Circle.     A  central  angle  is  the  angle 
between    two    radii.     Any    two   radii   form   two '  central 
angles,   which  are   distinguished   by  the    order   of  their 
lettering. 

An  inscribed  angle  is  an  angle  between  two  chords 
that  meet  on  the  circumference. 

The  arc  that  lies  between  the  arms  of  an  inscribed 
angle  or  a  central  a.ngle  is  said  to  be  subtended  by  the 
angle,  and  the  angle  is  said  to  stand  on  the  arc.  An  arc 
is  also  said  to  be  subtended  by  its  chord. 


SECTION   III.     CIRCLE   THEOREMS 

189.  Theorem  I.     In  the  same  circle,  or  in  equal  circles, 
if  two  central  angles  are  equal,  the  arcs  on  which  they 
stand  are  equal  also;   and  of  two  unequal  angles,  the 
greater  angle  stands  on  the  greater  arc. 

190.  COR.  1.     In  the  same  circle,  or  in  equal  circles,  if 
two  arcs  are  equal,  the  central  angles  standing  on  those 
arcs  are  equal  also ;  and  of  two  unequal  arcs,  the  greater 
has  the  greater  central  angle  standing  on  it. 

191.  COR.  2.     In  the  same  circle,  or  in  equal  circles, 
sectors  having  equal  central  angles  are  equal ;  and  of  two 
sectors  having  unequal  central  angles,  that  which  has  the 
greater  central  angle  is  the  greater. 

192.  COR.    3.      A  diameter  bisects  the  circle  and  its 
circumference. 

215.  An  arc  is  one  quarter  of  a  circumference  if  its  central  angle 
is  a  right  angle,  and  conversely. 

NOTE.     One  quarter  of  a  circumference  is  called  a  quadrant  of  arc. 

216.  An  arc  is  greater  than,  or  less  than,  a  quadrant,  if  its  central 
angle  is  greater  than,  or  less  than,  a  right  angle. 

193.  Semicircle;   Major  and  Minor  Arcs;  Complements, 
Supplements,  Explements.      One  half  a  circle  is  called  a 
semicircle,  and  one  half  a  circumference  is  called  a  semicir- 
cumference.      An   arc  less    than   a   semicircumference    is 
called  a  minor  arc;  one  greater  than  a  semicircumference 
is  called  a  major  arc. 

101 


102  CIRCLES 

Two  arcs  whose  sum  is  a  quadrant  are  called  comple- 
ments of  each  other ;  two  whose  sum  is  a  semicircumfer- 
ence  are  called  supplements  of  each  other ;  two  whose  sum 
is  a  circumference  are  called  explements  of  each  other. 

Notice  that  the  part  of  the  surface  of  a  circle  cut  off  by 
a  diameter  (a  semicircle)  is  a  segment,  and  also  a  sector. 

217.  If  two  arcs  are  equal,  their   complements  are  equal;   their 
supplements  are  equal ;  their  explements  are  equal. 

194.  Theorem  II.     In  the  same  circle,  or  in  equal  cir- 
cles, if  two  arcs  are  equal,  they  are  subtended  by  equal 
chords;  and  of  two  unequal  minor  arcs,  the  greater  is  sub- 
tended by  the  longer  chord. 

195.  COR.  1.     In  the  same  circle,  or  in  equal  circles,  if 
two  chords  are  equal,  they  subtend  equal  minor  and  equal 
major  arcs ;  and  of  two  unequal  chords,  the  longer  sub- 
tends the  greater  minor  arc. 

218.  If  two  equal  chords  of  a  circle  intersect,  they  include  a  pair 
of  equal  arcs. 

219.  An  inscribed  equilateral  polygon  is  regular. 

220.  An  inscribed  equilateral  hexagon  has  each  side  equal  to  a 
radius. 

221.  If  a  chord  equals  a  radius,  its  arc  is  one  sixth  of  the  circum- 
ference. 

196.  Theorem  III.     A  diameter  that  is  perpendicular 
to  a  chord  bisects  the  chord  and  its  subtended  arcs. 

197.  COR.  1.     A  diameter  that  bisects  a  chord  is  per- 
pendicular to  the  chord. 

198.  COR.  2.     The  perpendicular  bisector  of  a  chord 
passes  through  the  center  of  the  circle. 

222.  The  perpendicular  from  the  center  of  a  circle  to  a  side  of  an 
inscribed  equilateral  triangle  equals  one  half  the  radius. 


THEOREMS  103 

223.  The  sects  of  any  line  intercepted  between  the  circumferences 
of  two  circles  that  have  the  same  center  (concentric  circles')  are  equal. 

224.  Find  the  locus  of  the  center  of  a  circumference  that  passes 
through  two  given  points. 

225.  Find  the  chord  through  a  given  point  in  a  circle,  and  bisected 
by  that  point. 

226.  Find  the  locus  of  the  midpoints  of  parallel  chords. 

199.  Theorem   IV.     In  the  same  circle,   or  in  equal 
circles,  equal  chords  are  equidistant  from  the  center;  and 
of  two  unequal  chords,  the  longer  is  nearer  the  center. 

If  the  chords  are  placed  so  that  they  diverge  from  a 
common  point  on  the  circumference,  and  the  line  joining 
their  centers  is  drawn,  the  proof  can  be  easily  found.  Why 
is  this  position  a  good  one  in  which  to  place  the  chords  ?  . 

200.  COR.  1.     In  tlw  same  circle,  or  in  equal  circles, 
chords  that  are  equidistant  from  the  center  are  equal; 
and  of  two  chords  that  are  unequal  distances  from  the 
center,  the  one  nearer  the  center  is  the  longer. 

201.  COR.  2.     ji  diameter  of  a  circle  is  greater  than 
any  other  chord. 

227.  If  two  sects  from  a  given  point  to  a  given  circumference  are 
equal,  those  lines  are  equidistant  from  the  center  of  the  circle  ;  and  if 
the  lines  intersect  the  circumference  in  two  points  each,  the  sects  from 
the  given  point  to  the  second  points  on  the  circumference  are  also 
equal. 

228.  If  two  equal  chords  intersect,  the  sects  of  one  equal  the  sects 
of  the  other. 

229.  Find  the  locus  of  the  midpoints  of  the  equal  chords  of  a 
circle. 

230.  If  two  equal  chords  meet  at  a  point  on  a  circumference,  the 
bisector  of  the  angle  formed  by  them  is  a  diameter. 

231 .  If  two  circles  are  concentric,  all  chords  of  the  larger  which 
are  tangent  to  the  smaller  are  equal. 


104  CIRCLES 

202.  Ratio,  Proportion.     The  ratio  of  one  magnitude  to 
another  like  magnitude  is  that  number  by  which  the  sec- 
ond magnitude  must  be  multiplied  to  give  the  first  magni- 
tude.    For  example,  the  ratio  of  a  line  two  inches  long  to 
one  four  inches  long  is  one  half;  the  ratio  of  a  line  four 
feet  long  to  one  two  feet  long  is  two.     A  ratio  is  evidently 
a  quotient. 

If  the  ratio  of  one  pair  of  magnitudes  is  equal  to  the 
ratio  of  a  second  pair  of  magnitudes,  the  four  magni- 
tudes are  said  to  form  a  proportion,  and  are  said  to  be 
proportional.  The  four  magnitudes  used  in  a  proportion 
are  called  terms. 

WARNING.  It  needs  four  magnitudes  to  form  a  proportion.  Do  not 
say  that  two  things  are  proportional. 

203.  Measures:  Commensurable,  Incommensurable.    The 

measure  of  a  magnitude  is  its  ratio  to  a  unit  of  the  same 
kind  ;  as,  the  measure  of  a  line  is  its  ratio  to  an  inch,  a 
foot,  or  some  other  unit  of  length.  The  numerical  measure 
of  a  magnitude  is  its  ratio  to  a  unit,  expressed  numeric- 
ally ;  as,  five  feet. 

Two  magnitudes  are  commensurable,  if  some  third  magni- 
tude is  contained  a  whole  number  of  times  in  each,  with  no 
remainder. 

Two  magnitudes  are  incommensurable,  if  no  third  mag- 
nitude is  contained  evenly  in  both. 

The  ratio  of  two  like  commensurable  magnitudes  is  the 
ratio  of  the  numbers  of  times  some  third  magnitude  is 
contained  in  them  ;  as,  the  ratio  of  a  line  six  feet  long  to 
one  eleven  feet  long  is  six  to  eleven. 

NOTE.  Incommensurable  quantities  will  be  treated  in  the  Appen- 
dix, §  348. 


THEOREMS  105 

204.  Theorem  V.     In  the  same  circle,  or  in  equal  cir- 
cles, central  angles  are  proportional  to  the  arcs  on  which 
they  stand. 

The  commensurable  part  of  this  theorem  is  all  that  is  re- 
quired at  this  time;  that  is,  the  pupil  may  assume  that  the 
central  angles  are  commensurable  (have  a  common  divisor), 
and  so  show  that  the  ratio  obtained  for  them  will  be  th'e 
same  as  the  ratio  obtained  for  their  arcs.  Instead  of 
using  numbers  to  show  how  many  times  the  assumed  com- 
mon divisor  will  go  into  the  angles,  letters  should  be  used, 
so  that  the  proof  will  be  general.  For  the  incommensur- 
able discussion,  see  Appendix,  §  348. 

205.  COR.  1.     In  the  same  circle,  or  in  equal  circles, 
sectors  are  proportional  to  their  central  angles    (com. 
case  only). 

206.  Measurement   of    Central   Angles.     As    a   central 
angle  is  shown  by  Theorem  V  to  be  the  same  part  of  a 
perigon  that  its  arc  is  of  a  circle,  it  is  customary  to  say 
that  a  central  angle  is  measured  by  its  arc. 

The  reason  for  this  can  be  shown  more  clearly  by  the 
following  explanation  of  the  common  method  of  measuring 
angles : 

If  a  circumference  is  divided  into  360  equal  parts  called 
arc  degrees,  the  radii  joining  the  points  of  division  to  the 
center  will  divide  the  perigon  at  the  center  into  360  equal 
parts  called  degrees  of  angle;  also,  if  each  arc  degree  is 
divided  into  60  equal  parts,  called  minutes,  and  each  arc 
minute  into  60  equal  parts,  called  seconds,  the  radii  to 
these  points  of  division  will  form  minutes,  and  seconds,  of 
angle.  Then  if  any  arc  can  be  expressed  in  degrees, 
minutes,  and  seconds,  its  central  angle  will  evidently  be 


106  CIRCLES 

composed  of  the  same  numbers  of  degrees,  minutes,  and 
seconds  of  arc;   that  is,  the  angle  is  measured  by  its  arc. 

Theorem  V  might  then  have  been  stated:  A  central 
angle  is  measured  by  its  arc. 

207.  Theorem  VI.     din  angle  inscribed  in  a  circle  is 
measured  by  one  half  its  arc. 

To  make  this  theorem  complete,  it  must  be  proved  for 
an  inscribed  angle  having 

(1)  one  arm  a  diameter; 

(2)  one  arm  each  side  of  the  center; 

(3)  both  arms  the  same  side  of  the  center. 
Upon  what  must  this  theorem  depend  ? 

208.  COR.   1.     In  the  same  circle,  or  in  equal  circles, 
inscribed  angles  on  equal  arcs  are  equal ;  and  of  two  in- 
scribed angles  on  unequal  arcs,  that  on  the  larger  arc  is 
the  larger  angle,  and  conversely. 

209.  COR.   2.     (1)  An  angle  inscribed  in  a  semicircle 
is  a  right   angle.     (2)  An   angle  inscribed  on  an  arc 
greater  than,  or  less  than,  a  semicircumference,  is  gre-ater 
than,  or  less  than,  a  right  angle. 

232.  What  would  be  the  locus  of  the  vertices  of  all  triangles  hav- 
ing a  fixed  base  and  a  constant  vertex  angle? 

233.  Prove  that  the  bisectors  of  the  vertex  angles  of  all  triangles 
having  the  same  base  and  equal  vertex  angles  on  the  same  side  of 
that  base,  are  concurrent. 

234-  If  a  right  triangle  is  inscribed  in  a  circle,  its  hypotenuse  is 
the  diameter  of  the  circle. 

235.  A  circumference  described  on  any  side  of  a  triangle  as  a 
diameter  passes  through  the  feet  of  the  altitudes  to  the  other  sides. 
In  an  isosceles  triangle,  it  bisects  the  base  if  a  leg  is  used  as  diameter. 

236-  A  circumference  described  on  the  hypotenuse  of  a  right  tri- 
angle as  a  diameter  passes  through  the  vertex  of  the  right  angle. 


THEOREMS  107 

237.  Circumferences  described  on  two  sides  of  a  triangle  as  diam- 
eters intersect  on  the  third  side. 

238.  Three  consecutive  sides  of  an  inscribed  quadrilateral  subtend 
arcs  of  68°,  97°,  59°,  respectively.     Find  each  angle  of  the  quadrilat- 
eral, and  the  angles  between  the  diagonals. 

239.  Two  angles  of  an  inscribed  triangle  are  75°,  95°.     Find  the 
arcs  subtended  by  the  sides. 

210.  Theorem  VII.     An  angle  formed  by  a  tangent  and 
a  chord  of  a  circle  is  measured  by  one  half  the  arc  sub- 
tended by  the  chord.     Notice  that  there  are  two  angles 
between  the  chord  and  the  tangent,  and  that  each  is  meas- 
ured by  one  half  the  arc  on  the  same  side  of  the  chord  as 
the  angle  measured. 

211.  COR.  1.     Tangents  from  the  same  point  to  a  cir- 
cumference are  equal. 

212.  COR.   2.     Theline  joining  the  point  of  intersection 
of  two  tangents  to  the  center  of  tfo  circle  to  which  they  are 
tangent  bisects  the  angles  between  the  tangents,  and  the 
central  angle  between  the  radii  drawn  to  the  points  of 
contact. 

240.  Tangents  from  a  point  a  radius  distant  from  a  circle  make 
with  their  chord  of  contact  an  equilateral  triangle. 

241  •  The  sum  of  two  opposite  sides  of  a  circumscribed  quadrilat- 
eral equals  the  sum  of  the  other  two  sides. 

-  242.    A  parallelogram  circumscribed  about  a  circle  is  a  rhombus. 
243.   If  the  vertices  of  a  circumscribed  quadrilateral  are  joined  to 
the  center  of  the  circle,  the  non-adjacent  central  angles  formed  are 
supplemental. 

244-  The  angle  between  two  tangents  to  a  circle  is  double  the 
angle  between  the  chord  of  contact  and  the  radius"  to  a  point  of 
contact. 

245.  If  two  circles  are  tangent  to  each  other,  the  common  tangent 
at  their  point  of  tangency  bisects  the  other  common  tangents. 


108  CIRCLES 

213.  Theorem  VIII.     If  two  parallel  lines  meet  a  cir- 
cumference, the  arcs  cut  off  between  them  are  equal. 

246.  Lines  through  the  points  of  intersection  of  two  circles,  parallel 
to  each  other,  are  equal. 

247-  If  the  ends  of  two  unequal  parallel  chords  are  all  joined,  the 
points  of  intersection  and  the  center  of  the  circle  are  collinear. 

248.  If  the  ends  of  two  equal  arcs  of  a  circle  are  all  joined,  two 
of  the  lines  are  parallel,  the  other  two  cut  off  equal  sects  on  each  other, 

214.  Theorem  IX.     An  angle  formed  by   two  chords 
that  intersect  within  a  circle  is  measured  by  one  half  the 
sum  of  the  arcs  intercepted  by  the  arms  of  the  angle. 

215.  Theorem  X.     An  angle  formed  by  two  secants,  two 
tangents,  or  a  secant  and  a  tangent,  that  intersect  outside 
the  circle,  is  measured  by  one  half  the  difference  of  the 
intercepted  arcs. 

NOTE.  To  complete  the  discussion  of  angles  whose  arms,  or  arms 
extended,  meet  a  circumference,  the  measurement  of  the  supplements 
of  the  angles  mentioned  in  207,  210,  214,  and  215  should  be 
investigated. 

249.  An  exterior  angle  of   an  inscribed  quadrilateral  equals  the 
opposite  interior  angle. 

250.  The  sum  of  the  angles  inscribed  in  the  segments  of  a  circle 
cut  off  by  the  sides  of  an  inscribed  quadrilateral  equals  three  straight 


251.  The  ends  of  two  arcs  of  120°  and  60°  are  joined.     How  large 
are  the  angles  formed?     (Two  cases.) 

252.  The  opposite  sides  of  an  inscribed  quadrilateral,  no  two  of 
whose  sides  are  parallel,  are  extended  until  they  meet.      Find  the 
angle  between  the  bisectors  of  the  angles  formed. 

253.  If  two  chords  do  not  intersect,  and  the  midpoint  of  one  minor 
arc  of  one  chord  is  joined  to  the  ends  of  the  other  chord,  the  two 
triangles  formed  are  mutually  equiangular  (that  is,  the  angles  of  one 
are  equal  respectively  to  the  angles  of  the  other). 


THEOREMS  109 

$54.  If  three  angles  have  their  arms  pass  through  the  ends  of  the 
same  chord  of  a  circle,  their  vertices  being  on  the  same  side  of  the 
chord,  one  inside  the  circle,  one  on  the  circle,  the  other  outside  the  circle, 
the  angle  whose  vertex  is  outside  is  the  largest,  the  one  whose  vertex 
is  inside  is  the  smallest. 

216.  Theorem  XI.     The  opposite  angles  of  an  inscribed 
quadrilateral  are  supplemental. 

217.  COR.  1.     A  parallelogram  inscribed  in  a  circle  is 
a  rectangle. 

218.  COB.  2.     If  the  opposite  angles  of  a  quadrilateral 
are  supplemental,  the  quadrilateral  is  inscriptible. 

Through  how  many  vertices  can  a  circumference  be 
drawn  ?  What  if  the  other  fell  within  the  circle  ?  outside 
the  circle  ? 

Four  or  more  points  that  lie  on  the  same  circumference 
are  said  to  be  concyclic. 

255.  A  rectangle  is  inscriptible,  and  its  diagonals  intersect  at  the 
center. 

256.  In  an  inscribed  hexagon,  the  sum  of  three  angles,  no  two 
adjacent,  equals  the  sum  of  the  other  three.     What  general  statement 
for  polygons  of  an  even  number  of  sides  can  be  made? 

257.  A  circle  described  on  the  line  joining  the  orthocenter  of  a 
triangle  to  a  vertex  passes  through  the  feet  of  two  altitudes. 


SECTION  IV.    CONSTRUCTIONS  DEPENDING  ON  CIRCLE 
THEOREMS 

219.  CONST.  VIII.      To  bisect  a  given  arc. 

258.    Bisect  an  inscribed  angle,  without  using  its  arms. 
259-    Draw  a  tangent  to  a  given   arc  at  a  given   pointj  without 
using  the  center  of  the  circle. 

220.  CONST.  IX.      To  find  the  center  of  a  circle,  given 
any  arc. 

260.  Find  the  center  of  a  circle,  given  the  position  of  a  chord  and 
the  length  of  the  diameter. 

261.  Describe  a  circle,  given  the  positions  of  two  chords. 

221.  CONST.  X.      To  draw  a  tangent  from  a  given  point 
to  a  given  circle. 

262.  Draw  a  chord  of  a  given  circle,  through  a  given  point,  so  that 
it  will  have  a  given  length. 

263.  Draw  a  line  through  a  given  point,  so  that  its  distance  from 
a  given  point  may  equal  a  given  sect. 

264-  Construct  a  triangle,  given  the  base,  and  the  altitudes  to  the 
other  sides. 

265.  Construct  a  triangle,  given  the  base,  the  altitude  to  the  base, 
and  a  second  altitude. 

222.  CONST.  XI.      To  construct  a  circumference,  so  that 
a  given  angle  will  be  inscribed  on  the  arc  subtended  ly  a 
given  chord. 

What  other  kind  of  an  angle  is  measured  by  the  same 
arc  as  an  inscribed  angle  ?  Can  this  angle  be  constructed, 
given  the  chord  in  a  fixed  position  ?  How  can  the  center 
then  be  found  ? 

110 


CONSTRUCTIONS  USING  THEOREMS  111 

This  theorem  is  the  foundation  of  many  exercises  on 
loci.  The  most  common  way  of  using  this  as  an  example  of 
locus  is  :  "  Find  the  locus  of  the  vertex  of  a  given  angle  op- 
posite a  given  fixed  sect" 

When  a  locus  gives  a  circumference  for  its  result,  it  is 
because  of  one  of  two  things : 

(1)  because  the  point  in  question  is  always  at  a  fixed 
distance  from  a  given  fixed  point ; 

(2)  because  a  given  angle  is  always  opposite  a  given 
fixed   sect,  the  most  common  and  most   important   case 
being  that  of  a  right  angle  opposite  the  fixed  sect,  when 
the  locus  is  the  circumference  on  the  sect  as  diameter. 

266.  Construct  a  triangle,  given   the  base,  the  vertex  angle,  the 
median  to  the  base. 

267.  Construct  a  triangle,  given  the  base,  the  vertex   angle,  the 
altitude. 

268.  Construct  a  triangle,  given  the  base,  another  side,  the  angle 
between  the  median  to  the  base,  and  the  third  side. 

269.  Construct  a  triangle,  given  the  base,  the  altitude  to  the  base, 
the  angle  between  the  median  to  the  base,  and  a  side. 

270.  Find  the  locus  of  the  vertex  of  a  triangle,  given  the  base  and 
the  vertex  angle  ;  the  base  and  the  angle  between  the  median  and  a 
side. 

271.  Find  the  locus  of  the  midpoints  of  all  chords  whose  lines  pass 
through  a  given  fixed  point. 

272.  Find  the  locus  of  the  midpoint  of  a  sect  of  fixed  length, 
whose  ends  are  on  the  arms  of  a  given  fixed  right  angle. 

273.  Construct  a  parallelogram,  given  a  diagonal,  an  angle  opposite, 
the  angle  between  the  diagonals. 

274.  Construct  a  trapezoid,  given  the  bases,  a  side,  and  the  angle 
between  the  diagonals. 


112  CIRCLES 

223.     SUMMARY  OF  THEOREMS  AND  COROLLARIES,  BOOK  II 

(Numbers  in  parentheses  refer  to  black-faced  section  numbers.) 
I.   CIRCUMFERENCES  MEET  TWICE,   c.  s.  <sum  radii  and  >  ditf. 
(170). 

II.  CIRCLES  TANGENT.  c.  s.  =  sum  radii  or  =  diff.  (170) ; 
tangent  same  line  (173). 

III.  LINE  MEETS  A  CIRCUMFERENCE  TWICE.  Not  ±  radius  (167). 

IV.  LINE  TANGENT  TO  A  CIRCLE.     _L radius  at  end  (167). 

V.  LINES  EQUAL.  Center  1  in e_L  bisector  common  chord  (172)  ; 
chords  of  equal  arcs  (194)  ;  diameter  _L  chord  (106)  ;  =  chords  equi- 
distant from  center,  and  conversely  (199,  200)  ;  tangents  (211). 

VI.  LINES  UNEQUAL.  Chords  of  unequal  arcs  (194)  ;  ^  chords 
unequal  distances  from  the  center,  and  conversely  (199,  200);  diam- 
eter greatest  chord  (201). 

VII.   LINES  PERPENDICULAR.     Diameter  bisecting  chord  (197); 
Oinsc.  in  circle  (217). 

VIII.  LINE  PASSES  THROUGH  CENTER.  _L  bisector  chord  (198); 
_L  tangent  at  point  of  tangency  (167). 

IX.  ANGLES  EQUAL.  Central  angles  having  equal  arcs  (190); 
inscribed  angles  on  equal  arcs  (208) ;  angles  bisected  by  line  from 
center  to  intersection  of  tangents  (212). 

X.  ANGLES  UNEQUAL.  Central  angles  on  unequal  arcs  (190); 
inscribed  angles  on  unequal  arcs  (208,  209). 

XI.   ANGLES  SUPPLEMENTARY.     Opp.  angles  insc.  quad.  (216). 
XII.   ANGLES   MEASURED.     Central   angles,  by  their  arcs  (204, 
206). 

(a)    Angles  with  vertices  outside  the  circle,  by  one  half 

the  difference  of  the  intercepted  arcs  (215). 
(6)    Angles  with   vertices  on   the  circumference,  in- 
cluding inscribed  and  tangent  and  chord  angles, 
by  one  half  the  intercepted  arc :   an  angle  on  a 
semicircumference  equals  a  right  angle;  one  on 
more,  or  less  than,  a  semicircumference,  is  obtuse 
or  acute  (207,  210,  209). 
(c)    Angles  with  vertices  inside  the  circle,  by  one  half 

the  sum  of  the  intercepted  arcs  (214). 

XIII.  ARCS  EQUAL.  Equal  chords  (195)  ;  equal  central  angles 
(189)  ;  J.  bisector  chord  bisects  arc  (196)  ;  diam.  bisects  circumference 
(192);  equal  insc.  angles  (208)  ;  between  parallel  lines  (213). 


SUMMARY  OF  PROPOSITIONS  113 

XIV.   ARCS  UNEQUAL.    Unequal  central  angles  (189)  ;  unequal 
chords  (195);  unequal  inscribed  angles  (208). 

XV.    SECTORS  EQUAL.    Equal  central  angles  (191)  ;  diameter 
bisects  circle  (192). 

XVI.   SECTORS  UNEQUAL.     Unequal  central  angles  (191). 
XVII.    SECTORS  PROPORTIONAL.     To  their  central  angles  (205). 
XVIII.   QUADRILATERAL  INSCRIPTIBLE.    Opp.  angles  sup.  (218); 

CONSTRUCTIONS 

XIX.   LINE  PERPENDICULAR. 

(a)  Bisector  (180). 

(b)  At  point  in  the  line  (181). 

(c)  From  a  point  to  a  line  (181). 

(d)  At  the  end  of  the  sect,  without  extending  (182) . 
XX.  BISECT  ANGLE  (183). 

XXL  ANGLE  EQUAL  TO  A  GIVEN  ANGLE  (184). 

XXII.  PARALLEL  TO  A  GIVEN  LINE  (185). 

XXIII.  DIVIDE  SECT  INTO  EQUAL  PARTS  (186). 

XXIV.  BISECT  ARC  (219). 

XXV.   FIND  CENTER  OF  A  GIVEN  CIRCLE  (220). 
XXVI.   TANGENT  (221). 

XXVII.   CIRCUMFERENCE  HAVING  A  GIVEN  ANGLE  OPPOSITE  A 
GIVEN  FIXED  SECT  (222). 

LOCI 

XXVIII.   POINTS  EQUIDISTANT  FROM  A  FIXED  POINT  (179). 

XXIX.   VERTEX  OF  A  GIVEN  ANGLE  OPPOSITE  A  FIXED  SECT 
(222).     (See  XXVII.) 

224.  ORAL   AND  REVIEW   QUESTIONS 

Why  is  the  intersection  of  the  perpendicular  bisectors  of  the  sides 
of  a  triangle  called  the  circumcenter  ?  Why  are  the  intersections  of 
the  angle  bisectors  called  the  in-  and  ex-centers?  If  the  center  sect  of 
two  circles  is  10  ft.  long,  the  radii  being  8  ft.  and  4  ft.,  what  is  known 
about  the  meeting  of  circumferences  ?  For  what  length  center  sect, 
with  the  same  radii,  would  the  circles  be  internally  tangent  ?  When 
is  a  line  tangent  to  a  circle?  How  can  you  construct  a  tangent  at 
a  point  on  a  circle?  from  an  external  point  to  a  circle?  How  can 
SMITH'S  SYL.  PL.  GEOM.  —  8 


114  CIRCLES 

you  get  a  right  angle  opposite  a  given  sect?  any  angle?  Express  this 
as  a  locus.  Of  what  is  a  circumference  the  locus?  Define  chord. 
What  special  kind  of  chord  is  there?  Which  is  the  longest  chord 
of  a  circle?  Why?  WJien  are  arcs  equal?  central  angles  equal? 
inscribed  angles  equal?  How  can  you  measure  an  angle  with  its 
vertex  inside  the  circle?  its  vertex  on  the  circle?  its  vertex  outside 
the  circle  ?  What  kinds  of  angles  have  their  vertices  on  the  circumfer- 
ence? outside  the  circle?  What  is  known  about  the  angles  of  an 
inscribed  quadrilateral?  How  would  an  exterior  angle  of  the  quadri- 
lateral compare  with  the  opposite  interior  angle?  Why?  What 
principle  underlies  the  construction  of  perpendiculars?  Is  it  also 
used  for  the  proofs?  Why  does  the  perpendicular  bisector  of  a  chord 
pass  through  the  center?  Give  a  second  proof.  What  kind  of 
triangle  is  always  formed  by  two  radii  and  a  chord?  Can  you  tell 
any  proofs  where  this  fact  is  of  use?  What  is  the  key  to  drawing 
lines  parallel?  to  making  equal  angles?  to  dividing  a  sect  into  any 
number  of  equal  parts?  How  much  of  a  circle  is  needed  to  find 
its  center?  Are  lines  joining  the  ends  of  equal  arcs  always  parallel? 
What  is  known  about  all  equal  chords  of  a  circle  ?  What  is  the  locus 
of  the  midpoints  of  all  such  chords?  Upon  what  two  characteristics 
of  a  circle  do  the  circle  loci  depend?  How  many  points  need  to  be 
fixed  in  the  first?  in  the  second?  Explain  what  is  meant  by  a 
central  angle  being  measured  by  its  arc.  What  is  the  shortest  line 
from  a  point  to  a  circumference?  Why?  What  relation  has  the 
angle  between  two  tangents  to  the  angle  between  the  radii  to  the 
points  of  tangency  ?  What  two  ways  are  there  of  measuring  the  angle 
between  two  tangents?  If  the  arms  of  a  central  angle,  an  inscribed 
angle,  a  tangent  and  chord  angle,  are  all  respectively  parallel,  how  do 
their  arcs  compare?  Where  does  the  bisector  of  an  inscribed  angle 
meet  the  arc?  If  two  tangents  to  a  circle  are  parallel,  what  part  of 
the  circumference  is  between  them? 


GENERAL   EXERCISES 

275.  If  two  circles  are  tangent  internally,  and  the  radius  of  the 
larger  is  the  diameter  of  the  smaller,  any  chord  of  the  larger  drawn 
from  the  point  of  contact  is  bisected  by  the  smaller. 

276.  Construct  a  rhombus,  given  the  sum  of  the  diagonals  and  an 
angle. 


GENERAL  EXERCISES  115 

277.  Construct  a  right  triangle,  given  the  hypotenuse  and  the  sum 
of  the  other  sides. 

278.  An  inscribed  equiangular  polygon  of  odd  number  of  sides  is 
equilateral;   of  even  number  of  sides,   has  each   side  equal  to  the 
second  from  it. 

279.  The  tangents  at  the  vertices  of  an  inscribed  quadrilateral,  of 
which  two  opposite  angles  are  right  angles,  form  a  trapezoid. 

280.  Construct  a  rectangle,  given  the  difference  between  two  con- 
secutive sides  and  a  diagonal. 

281.  If  two  circles  intersect  twice,  the  angle  formed  by  joining  a 
point  of  intersection  to  the  ends  of  a  secant  through  the  other  point 
of  intersection  is  always  the  same. 

282.  If  the  feet  of  the  altitudes  of  a  triangle  are  joined,  the  angles 
of  the  triangle  formed  (called  the  pedal  triangle)  are  bisected  by  the 
altitudes. 

283.  Construct  a  triangle,  given  the  altitude,  a  base  angle,  the 
sect,  of  the  base  between  the  foot  of  the  altitude  and  the  second 

vertex. 

284~   Construct  a  triangle,  given  the  ba'se,  its  median,  a  base  angle. 

285.  Construct  a  rectangle,  given  a  diagonal  and  the  angle  between 
the  diagonals. 

286.  Construct  a  rectangle,  given  the  sum  of  two  consecutive  sides, 
and  the  angle  between  a  side  and  a  diagonal. 

287.  A  circumscribed  equiangular  polygon  is  equilateral. 

288.  A  circumscribed  equilateral  polygon  of  odd  number  of  sides 
is  regular,  and  has  its  sides  bisected  by  the  points  of  contact. 

289.  A  circumscribed  equilateral  polygon  of  even  number  of  sides 
has  its  alternate  angles  equal. 

290.  A  circumscribed  equilateral  triangle  has  each  altitude  equal 
to  three  radii. 

291.  Construct  an  isosceles  trapezoid,  given  the  longer  base,  the 
sum  of  altitude  and  side,  an  angle. 

292.  Construct  a  parallelogram,  given  the  sum  of  two  consecutive 
sides,  an  angle,  and  a  diagonal, 


116  CIRCLES 

293.  Construct  the  common  tangent  to  two  given  circles.     (Draw 
the  tangent,  the  radii  to  the  points  of  contact,  see  what  parts  of  the 
trapezoid  formed  are  given,  then  try  to  construct  it  from  those  given 
parts.     Notice  that  four  common  tangents  can  be  drawn  in  certain 
cases.) 

294.  If  two  circles  are  internally  tangent  at  P,  and  the  chord  AB 
of  the  larger  circle  is  tangent  to  the  smaller  circle  at  T,  prove  that  PT 
bisects  the  angle  APB. 

295.  If  triangle  ABC  is  circumscribed  about  a  circle  0,  with  CA 
and  A  B  tangent  at  fixed  points  T  and  T',  and  BC  any  third  tangent, 
then  CA  +  AB  —  BC  is  constant,  and  equals  the  diameter  if  Z.A  is  a 
right  angle. 

296.  If  two  parallel  tangents  are  crossed  by  a  third  tangent,  the 
third  tangent  subtends  a  right  angle  at  the  center. 

297.  In  295,  if  the  triangle  is  escribed,  AB  -f  BC  +  CA  is  con- 
stant. 

298.  Construct  a  trapezoid,  given  the  four  sides. 

299.  Construct  a  trapezoid,  given  the  parallel  bases  and  the  diago- 
nals. 

300.  Construct  a  square,  given  the  sum  of  a  diagonal  and  a  side. 

301.  Construct  a  square,  given  the  difference  of  a  diagonal  and  a 
side. 

302.  If  the  sum  of  two  opposite  sides  of  a  quadrilateral  equals  the 
sum   of  the  other  two  sides,  the  quadrilateral  can  be  circumscribed 
about  a  circle. 

303.  If  two  circles  intersect  in  two  points,  and  diameters  are  drawn 
from  one  of  those  points,  the  line  joining  the  other  ends  of  the  diame- 
ters is  double  the  center  sect,  and  passes  through  the  other  point  of 
intersection. 

304.  Through  a  point  within  a  circle,  draw  "the  chord  which  is 
bisected  by  that  point,  and  prove  that  it  is  the  shortest  chord  through 
the  point. 

305.  Describe  a  circle  tangent  to  two  given  lines,  to  one  at  a  given 
point. 


GENERAL   EXERCISES  117 

306.  Describe  a  circle  through  a  given  point,  touching  a,  given  line 
at  a  given  point. 

307.  Describe  a  circle  tangent  to  a  given  line  at  a  given  point,  and 
tangent  to  a  given  circle.     (Two  cases.) 

308.  In  a  circumscribed  cross  quadrilateral  the  difference  of  two 
opposite  sides  equals  the  difference  of  the  other  two  opposite  sides. 

309.  Find  the  locus  of  the  center  of  a  circle  which  has  a  given 
radius,  and  cuts  a  given  circle  at  the  ends  of  a  movable  diameter. 

310.  Find  the  locus  of  the  extremities  of  tangents  of  a  certain 
length  drawn  to  a  given  circle. 

311.  If  perpendiculars  are  drawn  from  the  ends  of  a  moving  diam- 
eter to  a  fixed  chord,  the  sum  of  those  perpendiculars  is  constant. 

312.  Construct  the  bisector  of  an  angle  without  using  its  vertex. 

313.  Construct  a  right  triangle,  given  the  two  sects  into  which  the 
bisector  of  the  right  angle  divides  the  hypotenuse. 

314-   Construct  a  right  triangle  having  a  given  fixed  hypotenuse, 
and  having  the  vertex  of  the  right  angle  in  a  given  fixed  line. 

315.  Construct  a  right  triangle,  having  a  fixed  hypotenuse,  and 
having  its  vertex  at  a  given  distance  from  a  given  point. 

316.  Construct  a  right  triangle,  having  a  fixed  hypotenuse,  and 
having  its  vertex  at  a  given  distance  from  a  given  line. 

317.  Draw  a  circle  with  a  given  radius,  tangent  to  two  given  circles. 

318.  Construct  a  right  triangle,  given  the  radius  of  the  inscribed 
circle  and  the  altitude  to  the  hypotenuse. 

319.  Draw  a  line  at  a  given  distance  from  two  given  points. 

320.  Construct  a  right  triangle,  given  the  radius  of  the  inscribed 
circle  and  the  bisector  of  the  right  angle. 

321.  Construct  a  right  triangle,  given  the  radius  of  the  circum- 
scribed circle,  and  (a)  an  acute  angle,  (6)  the  difference  of  the  acute 
angles. 

322.  Construct  a  triangle,  given  the  radius  of  the  circumscribed 
circle,  and  (a)  two  sides,  (6)  two  angles,  (c)  the  sects  of  the  base  made 
by  the  altitude. 


118  CIRCLES 

323.   Describe  a  circle  of  given  radius,  tangent  to  two  given  lines. 

324'  Describe  a  circle  tangent  to  two  given  lines  at  a  given  dis- 
tance from  their  intersection. 

325.  Draw  a  right  triangle,  given  the  altitude  and  the  bisector  of 
the  right  angle. 

326.  Construct  a  triangle,  given  a  side,  the  bisector  of  the  vertex 
angle,  the  sect  of  the  base  between  the  foot  of  the  altitude  and  the  end 
of  the  bisector  of  the  vertex  angle. 

327.  Construct  a  triangle,  given  the  sum  of  the  sides,  two  angles. 

328.  Construct  a  triangle,  given  the  sum  of  two  sides,  and  two 
angles. 

329.  Construct  a  triangle,  given  the  sum  of  two  sides,  one  of  the 
opposite  angles,  the  altitude  to  the  third  side. 

330.  Construct  a  triangle,  given  the  altitude  to  the  base,  a  base 
angle,  and  the  sect  from  that  vertex  to  the  foot  of  the  bisector  of  the 
vertex  angle. 

331.  Draw  a  circle  on  one  side  of  a  triangle  as  a  diameter,  and  so 
show  that  the  altitudes  from  the  ends  of  that  side  make  equal  angles 
with  the  other  sides  to  which  they  are  not  perpendicular. 

332.  If  two  circles  are  tangent  externally,  and  two  secants  are 
drawn  through  their  point  of  tangency,  the  chords  joining  the  ends  of 
the  secants  are  parallel. 

NOTE.  There  are  many  other  triangle  constructions  of  the  same 
kind  as  those  in  the  preceding  set,  and  the  pupil  can  make  exercises 
for  himself,  by  taking  any  three  parts  which  are  independent  (that  is, 
such  that  no  one  could  be  obtained  from  the  other  two)  and  trying  to 
form  the  triangle  from  those  parts.  Some  of  the  triangle  constructions 
cannot  be  done  until  the  fourth  book  has  been  studied. 

Notice  that  when  two  circles  are  used,  the  center  line  is  almost 
always  necessary  to  the  figure,  and  that  if  the  two  circles  intersect,  or 
are  tangent,  the  common  chord,  or  the  common  tangent  at  their 
point  of  tangency,  is  very  likely  to  be  needed. 


BOOK    III.     EQUIVALENCE   AND   AREA 

SECTION   I.     DEFINITIONS   AND  THEIR  DISCUSSION; 
FORMULAS 

225.  Equivalence.    Two  closed  figures  have  been  defined 
as  being  equivalent  when  they  contain  the  same  amount 
of  surface  (§  34).     The  question  at  once  arises  as  to  what 
ways  of  proving  figures  equivalent  are  known.     The  dis- 
tinction between  the  different  kinds  of  equality,  as  shown 
in  §§  32-37,  and   the   equality  axioms    (§  38),  give   the 
foundation  for  this  class ;  the  two  following  methods  show 
how  these  definitions  and  axioms  can  be  used. 

(1)  Congruent  figures  are  equivalent,  and  equivalent 
figures  added,  subtracted,  and  multiplied  or  divided  by 
the  same  number,  give  equivalent  results  (although  the 
results  need  not  be  congruent). 

(2)  The  whole  equals  the  sum  of  all  its  parts,  and  the 
other   equality  axioms   can   be   applied   to   the  equation 
obtained. 

The  equivalence  class  must  then  depend  on  these  two 
methods  :  the  equivalence  of  two  figures  usually  on  the 
first,  equivalence  equations  usually  on  the  second.  When 
the  first  is  used,  the  congruent  figures  are  often  added  or 
subtracted  in  different  positions,  thus  forming  new  figures 
which  are  equivalent,  but  not  congruent. 

226.  Addition  of  Polygons.     The  addition  of  two  poly- 
gons is  accomplished  by   placing   the   polygons   entirely 
outside  of  each  other,  as  regards  their  surface,  but  with 
some  portion  of  the  perimeter  (either  a  point,  or  some  one 

119 


120  EQUIVALENCE   AND  AREA 

or  more  sects)  in  common,  and  then  taking  the  whole 
figure  thus  formed.  The  common  part  of  the  boundary 
line  is  considered  as  omitted,  unless  it  is  a  point.  It  is 
evident  that  this  is  simply  the  ordinary  understanding  of 
a  sum,  the  only  difference  being  that  the  polygons  have 
to  be  placed  in  such  a  position  that  their  sum  can  be 
shown  as  a  single  figure. 

227.  Subtraction  of  Polygons.     The  subtraction  of  two 
polygons  is  accomplished  by  placing  the  smaller  polygon 
entirely  inside   the   larger,  and   then  omitting   the   part 
occupied  by  the  smaller ;  the  remaining  part  of  the  larger 
polygon  is  the  difference.     Here  again  the  usual  idea  of 
subtraction  is  employed,  the  figures  being  placed  so  that 
the  difference  shows  as  a  new  figure. 

228.  Addition  and  Subtraction  of  Parallelograms.     From 
the  definitions  of  addition  and  subtraction  of   polygons, 
it  can  be  seen  that  two  parallelograms  which  have  a  side 
and  an  angle  of  one  equal  to  a  side  and  an  angle  of  the 
other  can  be  added   or  subtracted   so  as  to  give  a  new 
parallelogram  having  the  same  side,  and  the  same  angle, 
as   the    result.     This   applies   to    rectangles   having  one 
equal  side,  and  the  fact  can  be  put  in  formal  statement  as 
follows : 

Two  rectangles  having  an  equal  side  can  be  added  so 
as  to  form  a  rectangle  having  the  equal  side  for  one  of 
its  sides,  the  sum  of  the  two  other  length  sides  of  the 
given  rectangles  for  its  other  side.  Similarly  for  the 
difference. 

229.  Rectangles  and  Squares.     A  rectangle  having  the 
sides  X  and  Y  is  spoken  of  as  the  n  X,  Y.     This  is  per- 
missible, since  all  rectangles  having  those  sides  are  con- 


FORMULAS  121 

gruent.     Similarly,  a  square  on  the  side  X  is  spoken  of 
as  the  D  X. 

Remembering  that  all  rectangles  which  have  two  sides 
equal  are  congruent,  it  is  easily  seen  that  all  that  it  is 
necessary  to  do  in  order  to  multiply  a  rectangle,  is  to 
multiply  one  side,  for  that  makes  a  new  rectangle,  composed 
of  as  many  congruent  rectangles  as  the  number  by  which 
the  side  was  multiplied.  This  is  most  easily  seen  by 
keeping  in  mind  that  multiplication  is  here  strictly  a  kind 
of  addition,  and  that  the  line  is  multiplied  by  continuing 
it  until  it  contains  the  required  number  of  equal  parts. 
In  exactly  the  same  way,  a  rectangle  can  be  divided  into 
any  number  of  equal  parts  by  dividing  one  side  into  that 
number  of  equal  parts,  and  so  forming  a  new  rectangle, 
having  one  side  the  required  part  of  the  given  side. 

NOTE.  Dividing  both  sides  of  a  rectangle,  or  multiplying  both 
sides,  performs  the  operation  on  the  rectangle  ttcice.  For  example, 
if  each  side  of  a  rectangle  is  doubled,  the  rectangle  is  made  four 
times  as  large. 

230.  Formulas.  By  the  use  of  the  axiom,  "the  whole 
equals  the  sum  of  all  its  parts,"  it  is  possible  to  obtain 
many  equations  between  rectangles  and  squares,  the  fol- 
lowing of  which  are  the  most  important: 

*(1)  The  square  on  the  sum  of  two  sects  is  equivalent 
to  the  sum  of  their  squares  plus  twice  their  rectangle. 

Take  two  given  sects,  4raw  the  square  on  their  sum, 
and.  see  if  it  contains  the  required  parts. 

*(2)  The  square  on  twice  a  sect  is  equivalent  to  four 
times  the  square  on  the  sect. 

*(3)  The  square  on  the  difference  of  two  sects  is  equiv- 
alent to  the  sum  of  their  squares,  less  twice  their  rec- 
tangle. 


122  EQUIVALENCE   AND   AREA 

Try  to  add  the  squares  of  the  sects,  and  subtract  the 
two  rectangles  in  such  a  manner  that  a  square  will  be  left. 

*(4)  The  difference  of  the  squares  on  two  sects  equals  a 
rectangle,  having  one  side  equal  to  the  sum  of  the  sects, 
the  other  side  equal  to  the  difference  of  the  sects. 

Subtract  the  smaller  square,  then  make  one  rectangle 
out  of  the  remainder. 

These  geometric  formulas,  which  deal  entirely  with  the 
surfaces  of  the  figures,  correspond  very  closely  to  certain 
algebraic  formulas ;  the  only  differences  between  the  two 
are  that  where  "  square  on  a  sect"  appears  in  the  Geometry, 
"square  of  a  number"  appears  in  Algebra,  and  where 
"rectangle  of  two  sects"  appears  in  Geometry,  "product 
of  two  numbers  "  appears  in  Algebra.  This  correspond- 
ence follows  throughout ;  wherever  an  algebraic  formula 
is  entirely  of  the  second  degree,  the  corresponding  geo- 
metric statement  concerning  squares  and  rectangles  is  also 
true.  The  reason  for  this  correspondence  will  be  evident 
from  the  third  theorem  and  its  corollaries. 

333.  Find  the  formula  for  the  square  on  the  sum  of  three  sects. 
Try  to  show  what  general  formula  can  be  obtained. 

334.  What  does  the  square  on  three  times  a  sect  equal?    What 
general  formula  is  there? 


SECTION   II.     THEOREMS 

231.  Theorem  I.     Parallelograms  (or  triangles)  on  the 
same  base,  or  on  equal  bases,  and  between  the  same  paral- 
lels, are  equivalent. 

Do  not  add,  for  this  method  does  not  work  for  all  posi- 
tions of  the  parallelograms. 

232.  COR.   1.       Parallelograms  (or  triangles)  on  the 
same  base,  or  on  equal  bases,  and  having  equal  altitudes, 
are  equivalent. 

Place  their  bases  on  the  same  line,  then  show  what  ? 

335.  Triangles  having  two  sides  equal,  and  the  included  angles 
supplemental,  are  equivalent. 

336.  Any  two  medians  of  a  triangle  form,  with  the  side  from 
whose  ends  they  are  drawn,  and  the  halves  of  the  sides  to  which  they 
are  drawn,  two  equivalent  triangles. 

337.  Any  median  divides  the  triangle  into  two  equivalent  triangles. 

838.   The  three  medians  of  a  triangle  divide  it  into  six  equivalent 
triangles. 

NOTE.     In  the  following  theorems,  it  is  the  surface  of  the  figure 
which  is  being  considered. 

233.  Theorem   II.     Two  rectangles  having  equal  alti- 
tudes are  proportional  to  their  bases.     (Commensurable 
case  only.) 

234.  COR.  1.     Two  rectangles  having  equal  bases  are 
proportional  to  their  altitudes. 

235.  Theorem  III.      The  ratio  betiveen  two  rectangles 

equals  the  product  of  the  ratio  of  their  bases  by  tJie  ratio 

123 


124  EQUIVALENCE   AND   AREA 

of  their  altitudes.  Can  their  ratio  be  found  at  once,  if 
they  have  no  equal  dimension?  To  what  could  the  ratio 
of  each  be  found  ? 

In  this  theorem,  and,  in  fact,  in  most  of  the  theorems 
of  this  book,  it  is  convenient  to  use  a  single  letter  to  rep- 
resent each  side  of  the  figure  rather  than  to  name  it  by  its 
end  points. 

236.  Area.     The  area  of  a  figure  has  been  defined  as 
the  amount  of  surface  inclosed,  and  this  area  is  usually 
stated  in'  terms  of  some  area  unit.     It  is  customary  to  call 
the  number  of  length,  or  linear  units  in  a  sect,  its  length; 
and  to  call  the  number  of  area  units  in  a  surface  its  area. 
The  unit  of  length  is  any  convenient  sect,  such  as  a  sect 
one  inch  long,  or  one  yard  long ;  the  unit  of  area  is  a 
square  having  a  linear  unit  for  a  side. 

In  dealing  with  area,  it  is  necessary  to  use  like  units ; 
that  is,  if  the  inch  is  used  as  a  length  unit,  then  the  square 
inch  must  be  used  as  the  area  unit,  and  so  for  all  units. 

To  find  the  area  of  a  figure,  then,  it  is  necessary  to  take 
the  ratio  of  the  figure  to  a  square  unit ;  while  to  find  the 
length  of  a  sect,  it  is  necessary  to  take  its  ratio  to  a  linear 
unit.  While  the  terms  "area"  and  "length"  will  be 
used  in  this  way,  the  student  should  bear  in  mind  that  it 
is  the  number  of  units  which  is  being  used,  both  in  area 
and  in  length. 

237.  Theorem  IV.     The  area  of  a  rectangle  equals  the 
product  of  its  base  by  its  altitude.     Read  §  236  carefully, 
then  take  the  ratio  of  the  given  rectangle  to  a  unit  of  area. 

238.  COR.  1.     The  area  of  any  parallelogram  equals 
the  product  of  its  base  by  its  altitude;  the  area  of  any  tri- 
angle equals  one  half  the  product  of  its  base  by  its  altitude. 


THEOREMS  125 

239.  COR.  2.     The  area  of  a  trapezoid  equals  one  half 
the  sum  of  its  bases  times  its  altitude. 

339.  The  area  of  a  trapezoid  equals  its  median  (the  line  joining 
the  midpoints  of  the  legs)  times  its  altitude. 

340.  The   area  of  a  circumscribed   polygon  equals   one  half   its 
perimeter  times  the  radius  of  the  circle. 

341.  A  line  through  the  midpoint  of  the  median  of  a  trapezoid, 
cutting  the  bases,  divides  the  trapezoid  into  two  equivalent  parts. 

240.  Theorem  V.     The  square  on  tlxe  hypotenuse  of  a 
right  triangle  is  equivalent  to  the  sum  of  the  squares  on 
the  other  two  sides. 

This  theorem  is  the  famous  Pythagorean  proposition,  so 
called  because  it  is  said  to  have  been  discovered  by 
Pythagoras.  It  is  probably  the  most  important  theorem 
in  Geometry  as  far  as  numerical  applications  are  concerned. 
It  has  been  claimed  that  about  350  proofs  for  this  theorem 
have  been  discovered ;  it  is  certain,  at  any  rate,  that  the 
number  of  proofs  is  very  great,  and  as  they  are  all  applica- 
tions of  the  equivalence  relations  that  have  been  given 
in  this  third  book,  the  student  ought  not  to  have  great 
difficulty  in  finding  several.  The  only  thing  to  be  avoided 
is  the  use  of  area  in  its  arithmetical  sense,  for  this  theorem 
must  be  proved  in  a  strictly  geometrical  sense,  and  so 
the  area  relations  (that  is,  the  formulas  for  the  area  of 
rectangles,  triangles,  etc.)  must  not  be  used.  Many  of 
the  proofs  are  obtained  by  adding  and  subtracting  equals, 
so  as  to  form  the  square  on  the  hypotenuse  in  one  case, 
and  the  sum  of  the  squares  on  the  other  sides  in  the  other 
case.  Probably  the  best  of  these  proofs  is  obtained  by 
drawing  the  square  on  the  hypotenuse  (so  as  not  to  include 
the  triangle)  and  completing  the  rectangle  through  its 
vertices,  using  the  legs  continued  as  two  sides  of  the  con- 


126  EQUIVALENCE   AND   AREA 

structed  rectangle.  This  new  figure  can  be  shown  to  have 
two  different  equivalence  values ;  one  is  obtained  from  the 
fact  that  it  can  be  proved  to  be  the  square  on  the  sum  of 
the  legs,  the  other  by  the  axiom  of  the  whole.  These  two 
values  will  quickly  give  the  required  relation  between  the 
squares. 

241.  COB.  1.     The  square  on  one  leg  of  aright  triangle 
is  equivalent  to  the  square  on  the  hypotenuse  less  the 
square  on  the  other  leg. 

242.  Numerical  Applications.     Since  the  area  of  a  figure 
is  the  number  of  units  in  the  figure,  many  of  the  equiva- 
lence  relations   can   be  used  for  numerical  applications. 
For  example,  the  area  of  a  triangle  in  square  units  can  be 
obtained  when  the  lengths  of  base  and  altitude  are  known 
in  length  units ;  as,  if  the  altitude  of  a  triangle  is  4  ft. 
long,  its  base  10  ft.  long,  then  the  area  of  the  triangle  is 
4  x  10  -v-  2  (or  20)  sq.  ft.     This  fact  applied  to  the  right 
triangle  formulas  enables  one  to  find  any  side  of  a  right 
triangle  if.  the  other  two  sides  are  known  ;  as,  in  a  right  tri- 
angle having  legs  3  ft.  long,  and  4  ft.  long,  respectively, 
the  hypotenuse  is  5  ft.  long,  for  the  area  of  a  square  is 
evidently  the  square  of  the  number  of  units  in  one  side, 
and  32  + 42=52.     Similarly,  if  the  hypotenuse  is  13  ft. 
long,  and  one  leg  is  12  ft.  long,  132  -  122  =  169  -  144  =  25 
=  52 ;  showing  that  the  other  leg  must  be  5  ft.  long. 

342.  Find  the  area  of  a  parallelogram  of  base  12  ft.  and  altitude 
14  ft.;  of  a  triangle  having  the  same  base  and  altitude. 

343.  In  order  that  a  triangle  should  be  equivalent  to  a  parallelo- 
gram of  the  same  base,  what  must  be  true  of  their  altitudes? 

344.  Find  the  area  of  a  trapezoid  of  bases  10  and  18  ft.,  altitude 
lift. 

345.  If  one  leg  of  a  right  triangle  is  3  ft.  long,  the  other  8  ft.  long, 
find  the  length  of  the  median  to  the  side  8. 


THEOREMS 


127 


846.  If  two  chords  of  a  circle  are  perpendicular  to  each  other,  the 
sum  of  the  squares  on  the  sects  formed  is  equivalent  to  the  square  on 
the  diameter. 

347.  In  right  triangle  ABC,  C  being  the  right  angle,  BD  is  drawn 
to  D  on  AC.  Prove  that  the  square  on  BD  plus  the  square  on  AC  is 
Equivalent  to  the  square  on  AB  plus  the  square  on  CD. 

348.  In  acute  triangle  ABC,  let  AB  =  7,  BC  =  S,  CV1=9;  call 
the  altitude  to  7,  li,  and  the  sect  from  A  to  the  foot  of  the  altitude,  p. 
Then  two  right  triangles  are  formed,  and  an  equation  can  be  written 
for  each.  Write  these  two  equations,  and  solve  them  for  h  and  p. 
Find  the  area  of  ABC.  Draw  the  median  to  AB  and  find  its  length 
from  the  new  right  triangle  formed. 

NOTE.  This  exercise  shows  that  the  use  of  the  theorem  for  the 
squares  on  the  sides  of  a  right  triangle  will  give,  when  the  lengths  of 
the  sides  of  a  triangle  are  known, 

(1)  the  length  of  each  altitude ; 

(2)  the  length  of  each  median ; 

(3)  the  lengths  of  the  sects  of  the  sides  made  by  the  altitudes ; 

(4)  the  area,  using  the  altitude  found. 

243.  Projections.  The  foot  of  the  perpendicular  from  a 
point  to  a  line  is  called  the  projection  of  that  point  on  the 
line.  If  both  ends  of  a  sect  are  projected  on  a  line,  the 
sect  between  the  projections  of  the  ends  of  the  given  sect 
is  called  the  projection  of  the  sect  on  the  line. 


P' 


B'     R 


S 


The  projection  of  P  is  P',  of  AB  is  AfBf,  of  US  is  RSr,  of 
KL  is  KfLf. 


128  EQUIVALENCE   AND   AREA 

244.  Theorem  VI.  The  square  on  a  side  of  an  oblique 
triangle  equals  the  sum  of  the  squares  on  the  other  two 
sides 

( 1 )  plus  twice  the  rectangle  of  either  of  those  sides  and 
the  projection  of  the  other  side  on  its  line,  if  the  opposite 
angle  is  obtuse  ; 

(2)  less  twice  the  rectangle  of  either  of  those  sides  and 
the  projection  of  the  other  side  on  its  line,  if  the  opposite 
angle  is  acute. 

The  acute  angle  case  will  have  to  be  proved  in  both  the 
acute  and  the  obtuse  angled  triangle.  In  what  kind  of  a 
triangle  is  the  square  on  a  side  known  ?  Then  what  must 
be  done  ? 

This  theorem  is  also  adaptable  to  numerical  work,  and 
if  the  letters  #,  J,  c,  stand  for  the  sides  of  a  triangle,  with 
a1  for  the  projection  of  a  on  5,  and  b'  for  the  projection  of 
b  on  «,  the  formula  obtained  would  be  (P  =  a2  +  b2.±  2a'b 
(or  2  «£>').  Note  that  if  three  of  these  parts  are  known, 
the  others  can  be  found,  and  that  the  kind  of  angle  oppo- 
site a  side  can  be  told  by  comparing  its  square  with  the 
sum  of  the  other  squares. 

849.  If  one  side  of  a  triangle  is  5,  the  projection  of  this  side  on  a 
second  side  is  2,  and  the  other  sect  of  that  side  is  4,  find  the  third 
side. 

350.  Does  Th.  VI  apply  to  right  triangles  also?     See  what  the 
projection  would  be  in  the  case  of  a  right  triangle.     Can  you  make  a 
general  statement  for  the  square  on  any  side  of  any  triangle,  regard- 
ing the  projection  as  negative  when  it  takes  off  from  a  side,  positive 
when  it  adds  to  the  side  ? 

351.  In  triangle  of  sides  a,  &,  c,  find  the  lengths  of  the  sects  of  c 
made  by  the  altitude  to  c. 

352.  Using  Th.  VI,  find  the  projection  of  b  on  c,  if  a  =  7,  b  =  10, 


THEOREMS  129 

245.  Area  and  Altitude  Formulas.  It  is  evident  that 
the  altitude  and  the  area  of  a  triangle  can  be  found  when 
the  three  sides  are  known,  either  by  the  right-triangle 
method,  or  by  the  use  of  the  formulas  for  the  square  on  a 
side  in  an  oblique  triangle.  If  the  altitude  and  area  are 
worked  out  for  a  triangle  of  sides  a,  5,  <?,  a  general  for- 
mula can  be  obtained,  which  can  be  applied  very  rapidly. 
If  h  is  the  altitude  to  <?,  bf  the  projection  of  b  on  c,  show 
that 

(l)^*^2-^-"2). 
(2)  h  = 


=  —  V(2  bo  +  b2  +  <?  -  a2)  (2  be  -  b2  -  &  +  a2), 


or     h  =    -V(a  +  b  +  C)(-a  +  b  +  c)  (a-l  +  c)  (a  +  5-c). 

(3)  Calling  the  sum  of  the  sides  2  s,  and  substituting, 
area  =  Vs  (s  —  a)  (s  —  b)  (s  —  c). 

APPLICATION.  If  the  sides  are  7,  8,  9,  then  s  (or  half 
the  sum)  =  12.  Subtracting  each  of  the  sides,  the  three 
remainders,  5,  4,  3,  are  obtained.  Then 

area  =  Vl2  x  5  x  4  x  3  =  12V5. 

To  find  any  altitude  by  this  method,  divide  the  area  by 
half  the  side  to  which  the  altitude  is  drawn  ;  for  example, 

24 

the  altitude  to  5  is  -r-V57 
o 

246.  The  Half  Equilateral  Triangle  ;  the  Isosceles  Right 
Triangle.  There  are  two  right  triangles  in  which  the 
relative  lengths  of  the  sides  can  be  told  at  once,  and  these 

SMITH'S  SYL.  PL.  GEOM.  —  9 


130  EQUIVALENCE   AND   AREA 

triangles  must  be  used  when  angles  are  given  to  find  the 
lengths  of  sides,  or  sides  are  given  to  find  the  size  of 
angles. 

If  an  altitude  is  drawn  in  an  equilateral  triangle,  two 
right  triangles  are  formed,  called  half  equilateral  trian- 
gles. The  altitude  meets  the  opposite  side  at  its  mid- 
point, so  one  leg  is  half  the  hypotenuse;  the  other  leg  is 
found  by  the  squares  on  the  sides.  The  altitude  is  always 
half  the  side  of  the  equilateral  triangle  times  the  square  root 
of  three;  and  the  area  of  the  equilateral  triangle  is  the  square 
of  half  the  side  times  the  square  root  of  three.  If  the  side 
is  10,  the  altitude  is  5V3^  and  the  area  is  25 V&  This 
triangle  has  angles  equal  to  one  third  a  straight  angle, 
and  one  sixth  a  straight  angle ;  or,  in  degrees,  of  60°  and 
30°. 

In  an  isosceles  right  triangle,  the  hypotenuse  is  a  leg 
times  the  square  root  of  two.  (Why  ?)  The  angles  here 
are  one  half  a  right  angle,  or  45°. 

853.  If  the  sides  of  a  triangle  are  29,  31,  50,  find  the  area  and  the 
altitude  to  50,  by  the  area  formula. 

354.  How  long  is  the  altitude  of  an  equilateral  triangle  of  side  14? 

355.  If  the  altitude  of  an  equilateral  triangle  is  20  V3,  find  the 
side. 

356.  If  the  area  of  an  equilateral  triangle  is  49  V3,  find  the  altitude. 

357.  If  the  altitude  of  an  equilateral  triangle  is  10,  find  the  side  ; 
if  the  area  is  10. 

358.  If  a  leg  of  an  isosceles  right  triangle  is  5,  find  the  hypotenuse. 

359.  If  two  thirds  of  one  side  of  an  equilateral  triangle,  and  one 
third  of  a  second  side,  are  cut  off  from  the  same  vertex,  the  line  join- 
ing those  points  is  perpendicular  to  the  side  from  which  one  third  is 
cut  off.     Does  this  need  to  be  one  third  ?    What  general  statement 
can  be  made? 

360.  If  the  vertex  angle  of  an  isosceles  triangle  is  30°,  and  each 
leg  is  10,  find  the  base ;  if  each  leg  is  a. 


THEOREMS  131 

361.  In  a  triangle,  having  an  angle  of  120°,  show  that  the  square 
on  the  opposite  side  equals  the  sum  of  the  squares  on  the  other  two 
sides,  plus  the  rectangle  of  those  sides. 

362.  In  a  triangle  having  one  angle  equal  to  one  third  of  a  straight 
angle,  the  square  on  the  opposite  side  equals  the  sum  of  the  squares 
on  the  other  two  sides  less  the  rectangle  of  those  sides. 

247.  Theorem  VII.      The  sum  of  the  squares  on  any 
two  sides  of  a  triangle  is  equivalent  to  twice  the  square 
on  one  half  the  third  side,  plus  twice  the  square  on  the 
median  to  that  side. 

The  previous  theorems  allow  us  to  take  but  one  square 
at  a  time ;  therefore,  for  many  purposes  this  theorem  is 
better,  for  it  takes  two  squares  at  once.  It  has  a  second 
purpose,  for  from  it  one  can  find  the  median  in  terms  of 
the  sides,  or  one  side  in  terms  of  the  other  sides  and  a 
median. 

248.  COR.  1.     The  difference  of  any  two  sides  of  a 
triangle  is  equivalent  to  twice  the  rectangle  of  the  third 
side  and  the  projection  of  its  median  on  it. 

363.  Find  the  median  to  5  in  the  triangle  of  sides  4,  5,  6. 

364'  Find  the  projection  of  the  median  to  5  on  that  line,  if  the 
sides  of  the  triangle  are  4,  5,  6. 

365.  If  the  median  to  a  side  of  a  triangle  is  7,  the  other  sides  are 
8  and  9,  find  the  side. 

366.  Show  that  the  sum  of  the  squares  on  the  sides  of  any  quadri- 
lateral equals  the  sum  of  the  squares  on  the  diagonals,  plus  four  times 
the  square   on  the  line  joining  their  midpoints.     What   does  this 
formula  become  if  the  quadrilateral  is  a  parallelogram? 

367.  The  sum  of  the  squares  on  the  diagonals  of  a  trapezoid  is 
equivalent  to  the  sum  of  the  squares  on  the  legs,  plus  twice  the  rec- 
tangle of  the  bases. 

368.  In  the  triangle  of  sides  a,  b,  c,  find  the  lengths  of  the  three 
medians. 


132  EQUIVALENCE   AND   AREA 

369.  Prove  that  the  sum  of  the  squares  on  the  medians  of  a  tri- 
angle equals  three  fourths  the  sum  of  the  squares  on  the  sides. 

370.  If  two  points  in  a  diameter  of  a  circle  are  equidistant  from 
the  center,  the  sum  of  the  squares  of  their  distances  from  a  moving 
point  on  the  circle  is  constant. 

NOTE.  The  number  of  numerical  exercises  founded  on  the  theo- 
rems of  this  book  is  practically  unlimited.  The  pupil  can  familiarize 
himself  with  many  of  the  different  kinds  by  taking  each  one  of  the 
formulas  and  assuming  values  for  all  but  one  of  the  sects  involved, 
and  solving  for  that  one.  For  example,  if  the  sides  of  a  triangle  are 

a,  by  c,  and  the  median  to  c  is  m,  then  a2  +  62  =  2  f -J  +  2m2;  assume 

values  for  (1)  a,  ft,  c,  (2)  a,  ft,  w,  (3)  a,  c,  m,  (4)  ft,  c,  m;  thus  using 
this  formula  in  all  possible  ways. 

In  assuming  lengths  for  lines  in  a  triangle,  care  must  be  taken 
that  the  figure  formed  is  possible ;  for  instance,  a  =  3,  b  =  7,  c  =  10  is 
not  possible.  Why  ? 

The  best  way  to  master  a  formula  is  to  investigate  each  way  it  can 
be  used,  so  that  no  question  about  it  can  be  entirely  unexpected. 


SECTION  III.     CONSTRUCTIONS 

249.  CONST.  I.      (1)   To  construct  a  polygon  equivalent 
to  a  given  polygon,  but  having  one  less  side. 

(2)   To  construct  a  triangle  equivalent  to  a  given  polygon. 

Note  that  any  two  consecutive  sides  add  to  the  surface 
only  a  triangular  surface,  which  could  equally  well  have 
been  contained  by  one  of  these  sides  and  the  extension  of 
one  of  the  sides  already  used  ;  in  other  words,  it  is  a  waste 
of  a  side  to  use  two  extra  sides  to  inclose  an  additional  trian- 
gular surface. 

250.  CONST.  II.      To  construct  a  rectangle  equivalent  to 
a  given  triangle. 

251.  CONST.  III.      To  construct  a  square  equivalent  to 
the  sum  of  two  given  squares. 

252.  COR.  1.      To  construct  a  square  equivalent  to 
the  difference  of  two  given  squares. 

253.  COR.  2.      To  construct  a  square  equivalent  to 
the  sum  of  any  number  of  given  squares. 

254.  COR.   3.      To   construct  a  square  equivalent  to 
any  whole  number  of  times  a  given  square. 

371.  Given  the  diagonals  of  two  squares,  construct  the  diagonal  of 
the  square  equal  to  the  sum  of  the  given  squares. 

372.  Construct  a  parallelogram  equivalent  to  a  given   triangle, 
having  its  diagonals  equal  to  two  sides  of  the  triangle. 

373.  Construct  a  rhombus  equivalent  to  a  given  triangle,  having 
one  of  its  sides  equal  to  a  side  of  the  triangle. 

133 


134  EQUIVALENCE  AND   AREA 

874-   Construct  a  rectangle  equivalent  to  any  given  polygon. 

375.  Construct   a   parallelogram  equivalent   to  a   given  triangle, 
having  one  side  equal  to  a  side  of  the  triangle,  and  having  a  given 
angle. 

376.  Construct  a  parallelogram  equivalent  to   a  given  triangle, 
having  its  two  sides  equal  to  two  of  the  sides  of  the  triangle. 

255.  CONST.  IV.      To  construct  a  square  equivalent  to  a 
given  rectangle. 

Any  of  the  formulas  that  give  equivalences  between 
squares  and  rectangles  could  be  used  to  make  this  con- 
struction, for  the  given  rectangle  could  represent  the  rec- 
tangle in  the  formula,  and  the  squares  could  then  be  added 
or  subtracted  as  stated  in  the  formula.  The  formula  for 
the  difference  of  two  squares  [§  230,  (4)]  is  probably  the 
best,  although  the  difference  of  the  squares  on  two  sides 
of  a  triangle  is  almost,  if  not  quite,  as  good.  If  the  first 
way  is  used,  what  must  the  sides  of  the  rectangle  be 
called  ?  Try  to  find  as  many  constructions  for  this  as 
possible,  for  they  are  excellent  practice  in  the  use  of  the 
equivalence  formulas;  there  are  at  least  seven  methods 
that  can  be  easily  seen  from  the  work  done  in  this  book. 

377.  Construct  a  rectangle   equivalent  to  a  given  square,  if  one 
side  of  the  rectangle  is  given. 

378.  Construct  a  rectangle  equivalent  to  a  given   square,  if  the 
perimeter  of  the  rectangle  is  given. 

379.  Construct  a  rectangle  equivalent  to  a  given  square,  if  the 
diagonal  of  the  rectangle  is  given. 

256,  SUMMARY  OF  THEOREMS   AND  COROLLARIES,  BOOK  III 

(Numbers  in  parentheses  refer  to  black-faced  section  numbers.) 
I.  PARALLELOGRAMS    EQUIVALENT.  =  bases,  bet.  ||s(231)  ;  =  bases 

and  =  alt.  (232);  prod,  bases  and  altitudes  =  (238). 

IT.  TRIANGLES  EQUIVALENT.  =  bases,  bet.  Us  (231)  ;  —  bases  and 

=  alt.  (232)  ;  prod,  bases  and  alt.  =  (238). 


SUMMARY  OF   PROPOSITIONS  135 

TIL  COMPARISON  OF  RECTANGLES.  Having  =  bases,  proportional 
to  alt.  (233)  ;  having  =  alt.,  proportional  to  bases  (234)  ;  proportional 
to  the  prod,  of  base  and  alt.  (235). 

IV.  AREAS.     (1)  rectangle  =  base  x  alt.  (237). 

(2)  Parallelogram  =  base  x  alt.  (238). 

(3)  Triangle  =  J  base  x  alt.    (238)  ;   equilateral,  =  the    square   of 
half  the  side,  times  >/3  (246). 

(4)  Trapezoid  —  \  sum  of  bases  x  alt.  (239). 

V.  EQUIVALENCE  FORMULAS  INVOLVING  SECTS  NOT  SIDES  OF  A 
TRIANGLE.     (1)  Square  on  sum  of  two  sects  equivalent  to  the  sum  of 
their  squares  plus  twice  their  rectangle  (230). 

(2)  Square  on  the  difference  of  two  sects  equivalent  to  the  sum  of 
their  squares  less  twice  their  rectangle  (230). 

(3)  Square  on   twice   a  sect  equivalent  to   four  squares   on  the 
sect  (230). 

(4)  Difference  of  the  squares  on  two  sects  equivalent  to  the  rec- 
tangle of  their  sum  and  their  difference  (230). 

VI.  SQUARE  ON  A  SIDE   OF   A    TRIANGLE.     General   Theorem:  — 
The  square  on  any  side  of  any  triangle  is  equivalent  to  the  sum  of  the 
squares  on  the  other  sides  plus  twice  the  rectangle  of  one  of  those 
sides  and  the  projection  of  the  other  on  its  line,  that  projection  being 
considered  positive  or  negative,  according  as  it  adds  to,  or  takes  away 
from,  the  side  on  which  it  is  projected. 

(1)  In  a  right  triangle,  square  on  hypotenuse  equivalent  to  the 
sum  of  the  squares  on  the  other  two  sides ;  square  on  a  leg  equivalent 
to  the  difference  of  the  squares  on  the  hypotenuse  and  the  other  leg 
(240,  241). 

(2)  Square  on  a  side  opposite  an  obtuse  angle  equivalent  to  the 
sum  of  the  squares  on  the  other  sides  plus  twice  the  rectangle  of  one 
of  those  sides  and  the  projection  of  the  other  on  it  (244). 

(3)  Square  on  a  side  opposite  an  acute  angle  equivalent  to  the  sum 
of  the  squares  on  the  other  sides  less  twice  the  rectangle  of  one  of 
those  sides  and  the  projection  of  the  other  on  it  (244). 

VII.  SQUARES  ON  Two  SIDES  OF  A  TRIANGLE.     (1)  Sum  equiva- 
lent to  twice  the  square  on  half  the  third  side  plus  twice  the  square 
on  the  median  to  that  side  (247). 

(2)  Difference  equivalent  to  twice  the  rectangle  of  the  third  side 
and  the  projection  of  its  median  on  it  (248). 


136  EQUIVALENCE   AND  AREA 


CONSTRUCTIONS 

VIII.  TRIANGLE,  equivalent  to  a  polygon  (249). 
IX.  RECTANGLE,  equivalent  to  a  triangle  (250). 
X.  SQUARE,  equivalent  to 

(1)  sum  of  two  squares  (251). 

(2)  difference  of  two  squares  (252). 

(3)  sum  of  any  number  of  squares  (253). 

(4)  any  number  times  a  square  (254). 

(5)  a  rectangle  (255). 

257.  ORAL  AND  REVIEW   QUESTIONS 

When  is  a  square  equivalent  to  the  sum  of  two  squares  ?  to  the  dif- 
ference? State  the  formula  for  the  square  on  a  side  of  a  triangle 
opposite  an  obtuse  angle ;  an  acute  angle.  Does  this  apply  to  right 
triangles?  Why?  What  is  the  method  for  finding  the  altitude  of 
an  equilateral  triangle  from  the  side  ?  the  area  from  the  side  ?  If  the 
side  is  24,  find  the  altitude;  the  area.  If  the  altitude  is  19 \/3,  what 
is  the  side?  If  the  area  is  9V3,  what  is  the  side?  If  the  side  is 
8V3,  how  far  is  it  from  the  centroid  to  a  vertex?  How  can  the  area 
of  a  triangle  be  most  easily  found  from  the  sides?  What  kind  of  a 
triangle  is  one  of  sides  3,  4,  5?  of  sides  9,  7,  5?  of  sides  7,  8,  9?  In 
what  way  can  the  sides  of  a  trapezoid  be  used  to  find  the  area  ?  Can 
the  sides  alone  of  a  parallelogram  be  used  to  find  the  area  t  Why  ? 
What  is  needed?  What  angles  can  be  found  from  the  lengths  of 
sides  of  triangles?  in  what  kinds  of  triangles?  What  are  the  rela- 
tive lengths  of  the  sides  of  a  triangle  having  angles  of  90°,  60°,  30°?  of 
90°,  45°,  45°?  If  a  leg  of  an  isosceles  triangle  were  known,  and  the 
vertex  angle  was  45°,  how  could  one  find  the  base  ?  if  the  vertex 
angle  were  30°  ?  In  constructing  a  triangle  equivalent  to  a  parallelo- 
gram, or  vice  versa,  what  relation  between  the  surfaces  should  be  kept 
in  mind?  Explain  in  a  few  words  the  general  method  used  in  reduc- 
ing a  polygon  to  a  triangle.  If  it  were  necessary  to  draw  a  square 
equivalent  to  a  given  polygon,  through  what  steps  would  it  be  neces- 
sary to  go  ?  a  square  equivalent  to  a  triangle  ?  to  a  parallelogram  ?  to 
the  sum  of  five  squares?  If  a  rectangle  were  to  be  constructed  equiv- 
alent to  a  given  square-,  and  having  a  given  side,  what  would  it  be 
best  to  call  the  side  of  the  rectangle?  Why?  What  three  theorems 


GENERAL   EXERCISES  137 

give  ratios  between  rectangles  ?  Are  all  likely  to  be  used  for  farther 
work,  or  have  some  served  their  purpose  ?  Why  was  an  extra  figure 
constructed  in  one  of  them?  What  does  area  mean?  When  a 
theorem  asks  for  the  ratio  of  two  like  magnitudes  for  the  first  time, 
what  must  be  done  with  those  magnitudes?  Then  in  terms  of  what 
is  the  ratio  expressed?  (NOTE.  The  commensurable  case  only  is 
being  considered.)  Which  formula  do  you  regard  as  likely  to  be 
used  the  most  frequently  of  all  those  in  this  book?  Why?  Explain 
two  methods  of  finding  the  projection  of  one  side  of  a  triangle  on 
another,  when  the  sides  are  known.  Explain  two  methods  of  finding 
the  area  of  a  triangle  from  the  sides.  Which  is  shorter?  Which 
can  be  most  readily  used  if  formulas  have  been  forgotten  ?  What  two 
methods  of  finding  the  median  from  the  sides  do  you  think  of? 
Which  is  shorter?  Which  can  be  used  if  formulas  are  forgotten? 
What  is  the  construction  for  finding  the  square  on  a  side  of  an  oblique 
triangle?  Why?  What  is  the  formula  for  the  area  of  a  rectangle? 
a  square?  a  parallelogram?  a  triangle?  an  equilateral  triangle?  a 
trapezoid  ? 

GENERAL   EXERCISES 

380.  If  a  square  field  and  a  rectangular  field  have  the  same  area, 
which  would  require  the  longer  fence? 

381.  Find  the  locus  of  a  point  such  that  the  squares  of  its  dis- 
tances from  two  fixed  points  shall  have  a  constant  sum.     (To  prove 
a  quantity  constant,  it  is  usually  best  to  show  that  it  equals  another 
quantity  which  is  constant.) 

382.  Find  the  locus  of  a  point  such  that  the  squares  of  its  distances 
from  two  fixed  points  shall  have  a  constant  difference. 

383.  If  any  point  within  a  rectangle  is  joined  to  the  vertices,  the 
sum  of  the  squares  of  the  sects  to  two  opposite  vertices  equals  the 
sum  of  the  squares  of  the  sects  to  the  other  vertices. 

384'  The  sum  of  tne  squares  on  the  diagonals  of  any  quadrilateral 
is  equivalent  to  twice  the  sum  of  the  squares  on  the  sects  joining  the 
midpoints  of  the  opposite  sides. 

385.  Construct  a  rectangle  equivalent  to  a  given  square,  if  the 
difference  of  the  sides  of  the  rectangle  is  given. 

386.  Find  the  three  altitudes  of  the  triangle  of  sides  a,  a,  b. 


138  EQUIVALENCE   AND   AREA 

387.  The  vertex  angle  of  an  isosceles  triangle  is  45°.  "  Find  the 
base,  if  the  leg  equals  10 ;  if  it  equals  a. 

388.  Find  the  area  of  a  trapezoid  of  bases  12,  18,  legs  4,  5. 

389.  Find  the  area  of  a  trapezoid  of  bases  a,  c,  legs  b,  d. 

390.  Find  the  side  of  an  equilateral  triangle  of  altitude  12. 

391.  If  the  altitude  of  an  equilateral  triangle  is  10  in.,  what  are 
the  radii  of  the  inscribed  and  circumscribed  circles?    if  the  side  is 
14  in.? 

392.  Compare  the  area  of  a  square  and  a  rhombus  of  the  same 
sides,  but  with  a  30°  angle;  a  60°  angle;  a  45°  angle. 

393.  The  sides  of  a  parallelogram  are  12  and  8.     Find  the  area 
if  an  angle  is  30°;  if  60°;  if  45°. 

394.  Construct  a  triangle  having  a  given  angle,  a  given  side  of  a 
certain  triangle,  and  equivalent  to  that  triangle. 

395.  Construct  a  parallelogram  equivalent  to  a  given  triangle, 
having  its  diagonals  equal  to  two  sides  of  the  triangle. 

396.  Find  the  diagonals  of  the  trapezoid  in  388. 


BOOK   IV.     SIMILAR   FIGURES. 
PROPORTIONS 

SECTION   I.     RATIO   AND   PROPORTION 

258.  Terms  of  a  Proportion.     Ratio  and  proportion  have 
already  been  defined  (§  202).     The  numerator  of  the  first 
ratio  and  the  denominator  of  the  second  ratio  (the  first 
and  last  terms)  are  called  extremes,  the  other  two  terms 
are  called  means.     The  numerators  of  the  two  ratios  are 
called  antecedents,  the  denominators  are  called  consequents. 
The  last  term  of  a  proportion  is  called  the  fourth  propor- 
tional. 

259.  Mean   Proportion.     If  the  means  of  a  proportion 
are  equal,  the  proportion  is  called  a  mean  proportion  ;  the 
mean  is  called  the  mean  proportional,  or  simply  the  mean, 
and  the  last  term  is  called  the  third  proportional. 

A  continued  proportion   is   a  series   of  equal  ratios  in 
which  any  two  successive  ratios  form  a  mean  proportion. 

260.  Proportion    Proofs.      In    proving    the    following 
proportion  theorems,  the  letters  a,  b,  c,  d,  may  be  used  to 
represent  like  geometrical  magnitudes  expressed  in  terms 
of  a  common  unit  of  measure.     Then,  while  the  ratio  may 
be  of   two  sects,  or  of   two   surfaces,  that   ratio  takes  a 
numerical  form  of  expression ;  asv  the  ratio  of  two  sects 
might  be  f . 

139 


140  PROPORTIONS 

These  proofs  must,  of  course,  depend  upon  the  equality 
axioms. 

261.  Composition  ;  Division.  Four  quantities  are  said 
to  be  in  proportion  by  composition  when  the  antecedents 
become  the  sums  of  the  terms  of  the  ratios  ;  as, 


b  d 

Four  quantities  are  said  to  be  in  proportion  by  division 
when  the  antecedents  become  the  differences  of  the  terms 

of  the  ratios;  as,          a-^^c^d,^ 

b  d 

If  the  ratios  of  the  sums  to  the  differences  are  used, 
the  quantities  are  said  to  be  in  proportion  by  composition 

a  +  b      c  -+-  d 


and  division;  as, 


a  —  b      c  —  d 


262.  Equimultiples.     Equimultiples   of   two   quantities 
are  the  results  obtained  by  multiplying  those  quantities  by 
the  same  number. 

NOTE.     In  proving  the  theorems  in  Ratio  and  Proportion,  use  the 
fractional  form  for  the  ratios,  and  use  the  proportion  as  an  equation  ;* 

as,  -  =  -•    In  this  way  the  equality  axioms  can  be  used  more  easily. 
b     d 

THEOREMS 

263.  Theorem  I.     If  four  quantities  are  in  proportion, 
the  product  of  their  means  equals  the  product  of  their 
extremes. 

264.  COB.  1.    In  a  mean  proportion,  the  square  of  the 
mean  equals  the  product  of  the  extremes. 

265.  COR.  2.    The  value  of  any  term  of  a  proportion 
can  be  expressed  in  the  other  terms  of  the  proportion. 


RATIO  AND  PROPORTION  141 

266.  Theorem  II.     If  the  product  of  two  quantities 
equals  the  product  of  two  other  quantities,  either  pair  can 
be  made  the  means,  and  the  other  pair  the  extremes,  of  a 
proportion. 

267.  COR.  1.    If  four  quantities  are  in  proportion,  they 
are  in  proportion  in  any  way  in  which  the  means  of  the 
given  proportion  are  either  both  means,  or  both  extremes, 
in  the  new  proportion. 

397.  Any  two  sides  of  a  triangle  are  inversely  proportional  to  the 
altitudes  drawn  to  them.  (Inversely  proportional  means  that  one  of 
the  ratios  is  inverted.)  Use  area  formula. 

268.  Theorem  III.     Four  quantities  which  are  in  pro- 
portion are  in  proportion  by  composition. 

269.  Theorem  IV.     Four  quantities  which  are  in  pro- 
portion are  in  proportion  by  division. 

270.  Theorem  V.     Four  quantities  which  are  in  pro- 
portion are  in  proportion  by  composition  and  division. 

271.  Theorem  VI.     If  four  quantities  are  in  proportion, 
equimultiples  of  the  antecedents  are  in  proportion  to  equi- 
multiples of  the  consequents. 

272.  Theorem  VII.     If  four  quantities  are  in  propor- 
tion, like  powers  of  those  quantities  are  in  proportion. 

273.  Theorem  VIII.     In  a  series  of  equal  ratios,  the 
ratio  of  the  sum  of  the  antecedents  to  the  sum  of  the  con- 
sequents equals  any  of  the  given  ratios. 

274.  SUMMARY 

If  a  proportion  is  written  in  fractional  form, 
and  the  four  terms  are  considered  as  forming  the  vertices  of  a  rec- 
tangle, its  terms  will  be  in  proportion  in  any  order  in  which  the  pairs 
are  taken  along  opposite  sides  of  the  rectangle,  in  the  same  direction  ; 


142  PROPORTIONS 

that  is,  both  to  the  right,  both  down,  etc.     For  example,  starting  from 

d  and  going  toward  the  left,  -  =  -. 

b      a 

Notice  that  the  proportion  never  goes  along  a  diagonal,  as  from 
a  to  d\  this  can  be  kept  in  mind  because  the  diagonals  form  a  multi- 
plication sign,  and  diagonal  terms  can  be  multiplied  but  not  divided. 

The  same  method  applies  to  composition  and  to  division,  the 
same  operations  being  applied  to  opposite  sides  to  form  the  new 
antecedents,  and  to  form  the  new  consequents.  For  example,  starting 

from  b  and  going  up,  ^±_^  =  *L±£. 
b  —  a      d  —  c 

This  method  can  be  extended  to  apply  to  equimultiples,  to  powers, 
and  to  composition  and  division  forms  involving  equimultiples  aiid 
powers,  and  in  this  way  it  serves  as  a  test  of  the  correctness  of  pro- 
portion forms. 

This  is  not  a  proof,  but  simply  a  test  for  correctness,  which  also 
acts  as  a  help  to  the  memory  by  combining  all  the  most  important 
proportion  forms  in  one  rule. 


SECTION  II.     PROPORTIONAL  SECTS 

275.  Pencil  of   Lines.     Lines  that  are  concurrent  are 
spoken  of  as  a  pencil  of  lines.     In  the  same  way  a  number 
of  lines  that  are  all  parallel  are  spoken  of  as  a  pencil  of 
parallels. 

276.  Theorem  I.     If  aline  is  cut  by  a  pencil  of  parallels, 
its  sects  are  proportional  to  the  sects  of  any  other  line  cut 
by  the  same  pencil  of  parallels,  including  as  a  special  case, 

A  line  parallel  to  the  base  of  a  triangle  cuts  the  sides, 
or  the  sides  extended,  so  that  the  sects  are  proportional. 
(Com.  case.) 

277.'  COR.  1.  A  line  parallel  to  the  base  of  a  triangle 
has  the  same  ratio  to  the  base  as  the  lengths  it  cuts  off  on 
the  other  sides  (from  their  common  vertex)  have  to  the 
whole  sides. 

278.  COR.  2.  If  parallel  lines  are  cut  by  a  pencil  of 
lines,  the  sects  cut  off  on  the  parallels  are  proportional. 

398.  In  the  quadrilateral  A  BCD,  having  angle  B  and  angle  D  right 
angles,  PE  and  PF  are  drawn  from  P  in  A  C  perpendicular  to  EC  and 
DA ,  respectively.     Prove  that  BE  :  E C  =  A  F :  FD. 

399.  If  EC,  of  triangle  ABC,  is  extended  to  Ar,  and  A  Y  is  cut  off 
on  AB  equal  to  CX,  then  XY  is  cut  by  CA  in  the  ratio  AB:BC. 

400.  The  diagonals  of  a  trapezoid  cut  each  other  proportionally, 
and  their  sects  are  proportional  to  the  bases. 

401.  If  aline  cuts  the  sides  of  a  triangle,  extended  if  necessary, 
the  product  of  three  non-consecutive  sects  equals  the  product  of  the 
other  three  sects. 

143 


144  PROPORTIONS 

279.  Points.  Cutting  a  Sect ;  Harmonic  Division.     A  point 
on  a  sect,  or  on  the  sect  extended,  is  said  to  cut  the  sect 
in  the  ratio  of  its  distances  from  the  ends  of  the  sect;  as, 
if  P  is  on  AB,  or  on  AB  extended,  the  ratio  in  which  it 
cuts  AB  is  PA :  PB.     If  the  point  is  three  fourths  as  far 
from  A  as  from  JS,  it  cuts  AB  in  the  ratio  3 :  4,  and  this  is 
the  same  whether  P  is  in  the  sect  itself  or  not. 

If  two  points  cut  the  same  sect  in  the  same  ratio,  one 
internally,  the  other  externally,  they  are  said  to  cut  the 
sect  harmonically.  The  equal  ratios  must,  of  course,  be 
taken  from  corresponding  ends  of  the  sect ;  as,  if  P  and  Q 
cut  AB  harmonically,  PA  :  PB  =  QA  :  QB. 

Notice  that  a  sect  cannot  be  cut  externally  in  the  ratio 
1,  for  if  P  is  in  the  extension  of  AB,  PA  cannot  equal  PB. 

280.  Theorem  II.     A  sect  can  be  cut  in  the  same  ratio 
internally  by  but  one  point,  and  externally  by  but  one 
point. 

402.  The  interior  common  tangents  of  two  circles  (those  between 
the  circles)  meet  the  center  line  at  the  same  point. 

403.  The  exterior  common  tangents  of  two  circles  meet  the  center 
line  at  the  same  point. 

404.  The  interior  and  exterior  common  tangents  to  two  circles  cut 
the  center  sect  harmonically. 

405.  A  line  through  the  ends  of  two  parallel  radii  of  two  circles 
meets  the  center  line  at  the  same  point  as  the  common  tangents. 

NOTE.  The  points  where  the  common  tangents  meet  the  center 
line  are  called  the  inverse,  and  direct  centers  of  similitude. 

406.  If  two  points  cut  a  sect  harmonically,  they  include  a  second 
sect,  which  is  cut  harmonically  by  the  ends  of  the  first  sect. 

281.  Theorem  III.     A  line  that   cuts  two  sides  of  a 
triangle  proportionally  is  parallel  to  the  third  side. 

407.  If  a  sect  joins  the  one  third  points  of  two  sides  of  a  triangle 
(taken  from  their  common  vertex),  what  part  of  the  third  side  is  it? 


SECTION  III.     SIMILAR  FIGURES 

282.  Similar  Figures.  Polygons  are  said  to  be  similar 
if  their  corresponding  angles  are  equal  and  their  corre- 
sponding sides  are  proportional.  See  also  Appendix,  §  349. 

There  are  now  three  things  which  can  be  proved  about 
polygons :  that  they  are  congruent,  equivalent,  or  similar. 
Equivalent  means  of  the  same  size  (as  regards  surface), 
similar  means  of  the  same  shape,  while  congruent  includes 
both  size  and  shape.  Notice  that  the  sign  for  congruent 
is  composed  of  the  equivalent  sign  and  the  similar  sign. 
These  facts  are  not  definitions  of  the  words,  but  serve  to 
show  the  distinction  in  meaning  in  a  somewhat  different 

light. 

*283.   Polygons  similar  to  the  same  polygon  are  similar 
to  each  other. 

*284.    Perimeters  of  similar  polygons  are  proportional  to 
any  pair  of  corresponding  sides. 

*285.    Regular  polygons  of  the  same  number  of  sides  are 
similar. 

408.  If  two  similar  polygons  are  placed  with  a  pair  of  correspond- 
ing sides  parallel  (the  polygons  lying  on  the  same  sides  of  those  lines), 
the  lines  through  the  corresponding  pairs  of  vertices  will  form  a  pen- 
cil, which  is  cut  proportionally  by  the  vertices  of  the  polygon. 

409.  If  two  polygons  lie  in  a  pencil  of  lines,  and  their  vertices  cut 
the  lines  proportionally,  the  polygons  are  similar. 

SMITH'S  SYL.  PL.  GEOM.  — 10          145 


146  SIMILAR   FIGURES 

286.  Theorem  IV.     Two  triangles  are  similar  if  two 
angles  of  one  are  equal  to  two  angles  of  the  other. 

410.  All  lines  through  the  point  of  tangency  of  two  circles  are  cut 
proportionally  by  the  circles. 

411.  If  AB  is  a  diameter  of  a  circle,  CD  a  chord  perpendicular  to 
AB,  then  any  chord  A  Y  cutting  CD  at  A'  has  the  product  AX  x  A  Y 
constant. 

412.  The  product  of  two  sides  of  a  triangle  equals  the  product  of 
the  altitude  to  the  third  side  by  the  diameter  of  the  circumscribed 
circle. 

413.  The  product  of  two  sides  of  a  triangle  equals  the  product  of 
the  bisector  of  the  included  angle  by  the  sect  of  thac  bisecting  line 
from  the  v,ertex  of  the  angle  to  the  circumscribed  circle. 

287.  Theorem  V.    Two  triangles  are  similar  if  two  sides 
of  one  are  proportional  to  two  sides  of  the  other,  and  the 
included  angles  are  equal. 

414.  In  any  triangle,  the  orthocenter,  the  centroid,  and  the  circum- 
center,  lie  in  a  straight  line,  and  the  distance  between  the  first  two  is 
double  the  distance  between  the  second  two. 

288.  Theorem  VI.      Two  triangles  are  similar  if  the 
sides  of  one  are  proportional  to  the  sides  of  the  other. 

If  one  had  the  included  angle  equal  to  that  of  the  other, 
the  triangles  would  be  similar ;  cut  it  off  equal,  and  show 
that  the  triangle  obtained  is  the  same  triangle  as  that 
given. 

289.  Similar  Triangles.     Similar  triangles  are  obtained 
much  as  congruent  triangles  were  obtained,  namely,  by 
three  parts.     The  angles  are  given  equal,  but  the  sides 
are  given  proportional  instead  of  equal.     Any  three  parts 
will  do,  except  two  sides  proportional  and  a  pair  of  angles, 
not  included,  equal.     The  following  table  will   serve  to 
show  the  relation  between  congruence  and  similarity. 


SIMILAR  FIGURES 


147 


GIVEN  PARTS      IF  SIDES  ABE  EQUAL      IF  SIDES  ARE  PROPORTIONAL 


3  sides : 

2  sides,  angle 

included, 
not  included, 


1  side,  2  angles 


3  angles : 


figures  congruent. 

figures  congruent ; 
figures  congruent    if 
angle  is  right  or  ob- 
tuse; or  if  acute,  with 
the  greater  side  oppo- 
site. Otherwise,  other 
angles   not   included 
are  supplemental, 
figures  congruent. 


figures  not  necessa- 
rily congruent. 


figures  similar. 

figures  similar, 
same  conditions  as 
for  congruent. 


figures  similar ;  side  not 
needed  as  two  sects  cannot 
be  proportional ;  same  as  3 
angles, 
figures  similar. 


Similar  figures,  and  especially  similar  triangles,  have 
many  practical  applications,  such  as  in  finding  the  height 
of  trees  and  buildings,  and  the  distances  between  objects. 
The  principle  involved  is  that  if  two  triangles  are  similar, 
any  one  side,  of  two  pairs  of  corresponding  sides,  can  be 
found  if  the  other  three  are  known.  Some  of  the  follow- 
ing exercises  illustrate  this  method. 

415.  How  tall  is  the  tree  that  casts  a  shadow  50  ft.  long  at  the 
same  time  that  a  pole  6  ft.  long  casts  a  shadow  8  ft.  long  ? 

416.  A  building  casts  a  shadow  64  ft.  long.     A  projection  on  one 
corner  of   the  building  that  is  found  to  be  8  ft.  from  the   ground 
casts  a  shadow  9  ft.  long.     How  high  is  the  building? 

417.  It  is  necessary  to  find  the  distance  from  B  to  an  inaccessible 
point  X.     A  line  BA  perpendicular  to  the  sighted  line  BX  is  laid  off 
146  ft.  long,  and  at  K,  60  ft.  from  A  on  A B,  a  perpendicular  to  AB 
is  erected,  meeting  the  sighted  line  AX  at  L.     If  KL  is  found  to  be 
30  ft.  long,  how  long  is  BX'! 


148  SIMILAR   FIGURES 

418.  Two  triangles  are  similar  if  the  sides  of  one  are  parallel  to 
the  sides  of  the  other. 

419.  Two  triangles  are  similar  if  the  sides  of  one  are  perpendicu- 
lar to  the  sides  of  the  other. 

420.  If  two  chords  of  a  circle  cut  each  other,  the  four  sects  are 
proportional. 

421'  If  two  secants  of  a  circle  intersect,  the  four  sects  from  the 
vertex  to  the  circle  are  proportional. 

NOTE.  There  are  now  two  principal  methods  by  which  to  find 
four  sects  proportional.  What  are  they? 

290.  Theorem  VII.     Tlw  areas  of  triangles,  or  of  paral- 
lelograms, having  an  angle  of  one  equal  to  an  angle  of  the 
other,  have  the  same  ratio  as  the  product  of  the  sides  in- 
cluding that  angle. 

291.  COR.  1.     The  areas  of  similar  triangles  have  the 
same  ratio  as  the  squares  of  their  corresponding  sides. 

292.  COR.  2.     If  two  triangles  that  have  an  angle  of 
one  equal  to  an  angle  of  the  other  are  equivalent,  the  prod- 
uct of  the  sides  including  the  angle  in  one  equals  the 
product  of  the  sides  including  the  angle  in  tlxe  other, 
and  conversely. 

422.  If  BC,  of  triangle  ABC,  is  8,  and  CA  is  6,  how  long  must 
CFbe,  so  that  a  line  from  F,  on  BC,  to  X,  the  two  thirds  point  of 
CA,  will  cut  the  triangle  into  equivalent  parts. 

423.  A  line  from  the  midpoint  of  a  side  of  a  triangle  must  go  to 
what  point  on  a  second  side  to  form  an  equivalent  triangle? 

424.  In  a  right  triangle  of  legs  3  and  4  ft.,  the  hypotenuse  is 
extended  10  ft.      How  long  must  a  leg  be  extended  at  the  same 
vertex  so  that  the  line  joining  the  extremities  of  the  extensions  will 
form  a  triangle  double  the  given  triangle?     (Two  cases.) 

4%5.  If  similar  triangles  are  drawn  upon  the  sides  of  a  right 
triangle  as  corresponding  sides,  the  triangle  on  the  hypotenuse  equals 
the  sum  of  the  other  triangles. 


SIMILAR  FIGURES  149 

293.  Theorem  VIII.     //  two  polygons  are  similar,  they 
can  be  divided  into  the  same  number  of  triangles,  simi- 
lar each  to  each,  and  similarly  placed. 

294.  Theorem  IX.     If  two  polygons  are  composed  of 
the  same  number  of  triangles,  similar  each   to  each, 
and  similarly  placed,  the  polygons  are  similar. 

295.  Theorem  X.     TJw  areas  of  similar  polygons  have 
the  same  ratio  as  the  squares  of  any  pair  of  correspond- 
ing sides. 

426.  If  similar  polygons  are  constructed  on  the  sides  of  a  right 
triangle  as  corresponding  sides,  the  polygon  on  the  hypotenuse  equals 
the  sum  of  the  other  two  polygons. 

427.  If  each  side  of  one  polygon  is  double  the  corresponding  side 
of  a  second  similar  polygon,  what  relation  have  the  areas? 

428.  If  the  area  of  one  polygon  is  36  times  the  area  of  a  similar 
polygon,  what  relation  have  the  sides? 

NOTE.  Area  ratios  in  similar  figures  are  always  the  squares  of  line 
ratios;  line  ratios  are  the  square  roots  of  area  ratios. 

296.  Theorem  XI.     In  a  right  triangle,  the  altitude 
to  the  hypotenuse  divides  the  triangle  into  two  triangles 
similar  to  each  other,  and  to  the  whole  triangle. 

297.  COR.  1.     (1)  The  altitude  to  the  hypotenuse  is  the 
mean  proportional  between  the  sects  of  the  hypotenuse. 

(2)  Either  leg  of  the  triangle  is  the  mean  propor- 
tional between  the  hypotenuse  and  its  own  projection  on 
the  hypotenuse. 

429.  In  a  right  triangle  of  legs  5  and  12,  find  the  projections  of  5 
and  of  12,  on  the  hypotenuse;  find  the  altitude  to  the  hypotenuse. 

430.  In  a  right  triangle  of  legs  «,  />,  hypotenuse  c,  find  the  projec- 
tions of  a  and  of  b  on  c ;  find  the  altitude  to  c. 

NOTE.  The  results  of  exercise  430  are  formulas  that  hold  for  all 
right  triangles. 


150  SIMILAR   FIGUllKS 

431.  Tangents  from  a  point  to  a  circle  of  radius  6  are  of  length 
8.    Find  the  chord  of  contact. 

432.  In  a  right  triangle,  the  sects  of  the  hypotenuse  made  by  the 
altitude  are  4  and  5.     Find  the  other  sides  and  the  altitude. 

433.  The  squares  of  two  chords  drawn  from  a  point  on  a  circle 
have  the  same  ratio  as  their  projections  on  the  diameter  from  that 
point. 

434.  The  half  chord  perpendicular  to  a  diameter  is  the   mean 
between  the  sects  of  the  diameter. 

298.  Theorem  XII.     A    line   that    bisects    an    angle, 
interior  or  exterior,  of  a  triangle,  divides  the  opposite 
side,  internally  or  externally,  into  sects  proportional  to 
the  other  sides  of  the  triangle.     Note  that  the  two  lines 
divide  harmonically. 

435'  If  a  line  drawn  from  the  vertex  of  an  angle  of  a  triangle 
divides  the  opposite  side,  internally  or  externally,  in  the  ratio  of  the 
other  two  sides,  the  line  bisects  the  angle,  interior  or  exterior,  from 
whose  vertex  it  is  drawn. 

436.  In  a  triangle  of  sides  6,  7,  8,  find  the  sects  of  7  made  by  the 
bisector  of  the  opposite  angle ;  of  the  exterior  angle  at  the  opposite 
vertex. 

437.  In  a  triangle  of  sides  a,  b,  c,  find  the  sects  of  c  made  by  the 
bisector  of  the  opposite  angle ;  of  the  exterior  angle  at  the  opposite 
vertex. 

NOTE.      The  results  of  437  are  formulas  that  hold  for  all  triangles. 

438.  If  two  lines  from  the  vertex  of  the  right  angle  of  a  right 
triangle  make  equal  angles  with  one  of  the  legs,  they  cut  the  hypote- 
nuse harmonically. 

299.  Theorem  XIII.     If  a  pencil  of  lines  cuts  a  cir- 
cumference,  the   lines   are   cut   proportionally,   so   that 
the  product  of  the  two  sects  from  the  vertex  to  the  cir- 
cumference on  one  line  is  the  same  as  that  product  on 
any  other  line. 


SIMILAR  FIGURES  151 

300.  COR.  1.  A  tangent  from  the  vertex  of  a  pencil 
to  a  circumference  is  the  inean  proportional  between 
the  two  sects,  from  the  vertex  to  the  circumference, 
of  any  other  line  of  the  pencil  that '  is  cut  by  the 
circumference. 

439.  If  two  points  are  taken  on  each  line  of  a  pencil,  so  that  the 
product  of  the  two  sects  from  the  vertex  is  the  same  for  all  the  lines 
of  the  pencil,  the  two  points  being  on  opposite  sides  of  the  vertex, 
the  four  points  on  any  two  lines  are  concyclic. 

440.  In  the  figure  of  439,  if  the  two  points  on  each  line  are  on 
the  same  side  of  the  vertex,  the  four  points  on  any  two  lines  are 
concyclic. 

441.  Find  the  locus  of  the  point  P  on  a  secant  to  a  given  circle 
which  cuts  the  circle  in  changing  points,  A  and  B,  so  that  PA  x  PB 
is  constant. 

442-  If  two  intersecting  lines  are  cut,  one  by  one  point,  the  other 
by  two  points  on  the  same  side  of  the  vertex,  so  that  the  sect  cut  off 
by  the  one  point  is  the  mean  proportional  between  the  sects  from  the 
vertex  on  the  other  line,  a  circle  through  the  three  points  would  be 
cangent  to  the  line  cut  by  the  one  point. 

443.  Construct  a  circle  through  two  given  points  tangent  to  a 
given  line. 

444.  The  product  of  two  sides  of  a  triangle  equals  the  square  of 
the  bisector  of  the  included  angle,  plus  the  product  of  the  sects  into 
which  the  bisector  divides  the  opposite  side. 

SUGGESTION.     Inscribe  the  triangle. 

445-   If  three  circles  intersect,  their  common  chords  are  concurrent. 

446.  If  two  circles  intersect,  tangents  from  any  point  in  their 
common  chord  extended,  to  the  two  circles,  are  equal. 


SECTION   IV.     CONSTRUCTIONS 

301.  CONST.   I.      To  divide  a  given  sect  internally,  and 
externally,  in  a  given  ratio  (or,  harmonically). 

NOTE.   A  given  ratio  is  always  represented  by  two  given  sects,  the 
given  ratio  being  that  of  those  sects. 

302.  COR.  1.     To  divide  a  given  sect  into  parts  propor- 
tional to  any  number  of  given  sects. 

447.  Given  a  sect,  and  one  point  of  harmonic  division,  find  the 
other. 

448-  Given  a  sect,  cut  it  into  parts  having  the  ratio  1:2:5. 

303.  CONST.  II.      To  find  the  fourth  proportional  to  three 
given  sects. 

304.  COR.  1.     To  find  the  third  proportional  to  three 
given  sects. 

449>   Draw  a  line  through  a  given  point  so  that  it  will  cut  off  sects 
having  a  given  ratio  on  the  arms  of  a  given  angle. 

450.  Draw  a  line  from  a  given  point  to  a  given  line,  so  that  it  will 
have  a  given  ratio  to  the  perpendicular  from  that  point. 

451.  Draw  a  line  through  a  given  point  so  that  it  will  be  cut  in  a 
given  ratio  by  the  arms  of  the  angle. 

452-    Construct  two  sects,  given  their  sum  and  their  ratio. 
453.   Construct  two  sects,  given  their  difference  and  their  ratio. 

305.  CONST.    III.      To  find  the  mean  proportional  between 
two  given  sects.     Find  three  methods. 

This  construction  is  the  foundation  of  all  square  root 
questions  in  the  Geometry.     Since  the  square  of  the  mean 

152 


CONSTRUCTIONS  153 

equals  the  product  of  the  extremes,  it  follows  that  the 
mean  equals  the  square  root  of  the  product  of  the  extremes. 
It  is,  therefore,  possible  to  construct  the  square  root  of 
any  required  number  by  making  the  extremes  of  such 
length  that  their  product  equals  the  number  of  which  the 
square  root  is  asked.  This  is  used  also  in  constructing 
figures  whose  areas  have  a  certain  ratio,  for  if  the  figures 
are  similar,  the  line  ratio  is  the  square  root  of  the  given 
area  ratio,  and  so  can  be  found  by  the  mean  proportional. 

454.  Construct  a  sect  V2in.  long,  given  the  sect  1  in.  long. 

455.  Explain  how  the  sect   Vn  in.  long  could  be  constructed  for 
any  value  of  n,  given  a  sect  of  1  in. 

456.  Find  another  way  to  construct  the  square  root  of  a  number. 

306.  Mean  and  Extreme  Ratio.  If  a  sect  is  divided  so 
that  the  longer  part  is  the  mean  between  the  whole  sect 
and  the  shorter  part,  the  sect  is  said  to  be  divided  in 
mean  and  extreme  ratio.  If  AB  is  divided  by  P  so  that 

• — = — -,  then  P  divides  AB  in  mean  and  extreme  ratio. 
AP  PB 

There  is  an  external  mean  and  extreme  division  as  well  as 
an  internal ;  the  words  "  mean  "  and  "  extreme  "  do  not 
refer  to  the  interior  and  exterior  cases,  but  to  the  position 
in  which  the  parts  occur  in  the  proportion. 

It  should  be  noticed  that,  as  the  proportion  stands,  there 
are  two  unknown  sects  used  ;  by  a  proper  transforming  of 
the  proportion  the  parts  can  be  combined  (in  different 
ways  for  the  two  cases)  so  that  the  given  sect  AB  will 
appear  more  often,  thus  displacing  the  unknowns.  The 
new  form,  in  which  the  unknowns  occur  as  seldom  as 
possible,  is  the  one  with  which  to  work  in  attempting 
the  construction;  it  will  be  found  very  easy  if  attacked 
logically. 


154  PKOPORTIONS 

307.  CONST.  IV.      To  divide  a  given  sect  in  mean  and 
extreme  ratio. 

437.  If  a  line  10  in.  long  is  divided  in  mean  and  extreme  ratio, 
how  long  is  the  mean  sect? 

458.  If  the  bisector  of  a  base  angle  of  an  isosceles  triangle  cuts  off 
the  longer  sect  of  the  opposite  side  equal  to  the  shorter,  of  the  sides 
including  the  bisected  angle,  that  side  is  cut  in  mean  and  extreme 
ratio. 

308.  CONST.  V.      To  construct   a  polygon  similar  to  a 
given  polygon,  on  a  given  sect  as  a  side  corresponding  to  a 
certain  side  of  the  given  polygon. 

459.  Construct  a  polygon  similar  to  a  given  polygon  and  having 
twice  as  great  an  area;  n  times  as  great  an  area  (n  a  whole  number). 

460.  Construct  a  polygon  similar  to  a  given  polygon  and  having 
an  area  one  half  as  great;  three  fifths  as  great;   —  times  as  great. 

461.  Given  two  similar  polygons,  construct  a  polygon  similar  to 
them  and  equal  to  their  sum. 

462.  Given  two  similar  polygons,  construct  a  polygon  similar  to 
them  and  equal  to  their  difference. 

309.  -  CONST.  VI.     (1)    To  draw   a  rectangle   having  a 
given    ratio    to    a  given    square.     (2)    To    draw  a   square 
having  a  given  ratio  to  a  given  square. 

Notice  that  the  construction  "  To  draw  a  square  equiva- 
lent to  a  given  rectangle  "  uses  a  method  of  construction 
which  is  the  same  as  drawing  a  mean  proportional. 
Why? 

463.  Construct  a  square  three  fourths  as  large  as  a  given  square. 

464.  Construct   a   square  five  times  as  large  as  a  given   square 
by  the  method  of  §  309. 

310.  SUMMARY  OF  THEOREMS  AND  COROLLARIES,  BOOK  IV 

(Numbers  in  parentheses  refer  to  black-faced  section  numbers.) 
I.    SECTS  PROPORTIONAL.     If  cut  off  on  transversals  by  Us  (270) ; 
cut  off  on  Us  by  a  pencil  (278)  ;  sides  of  A  cut  by  a  II  to  third  side  (276) ; 


SUMMARY  OF   PROPOSITIONS  155 

line  parallel  to  side  of  A  has  the  same  ratio  to  it  as  the  length  cut 
off  on  another  side  has  to  that  side  (277)  ;  sides  of  ~  polygons  (282)  ; 
perimeters  of  ~  polygons  proportional  to  sides  (284);  in  a  right 
triangle  the  altitude  is  the  mean,  either  leg  is  a  mean  (297)  ;  line 
bisecting  an  angle  of  a  A  divides  the  opposite  side  (298) ;  pencil 
cutting  a  circle,  the  product  of  the  sects  is  constant  (299);  the 
tangent  is  the  mean  (300). 

II.  SECT   CUT.     By  but  one  internal  point  in  a  given  ratio,  by 
but  one  external  point  in  a  given  ratio  (280). 

III.  LINES  PARALLEL.      Line  cutting  two    sides    of  A   propor- 
tionally (281). 

IV.  FIGURES  ARE  SIMILAR.     ~  to  the  same  polygon  (283);  regu- 
lar, same  number  of  sides  (285) ;  A,  if  they  have  2  A  =  (286),  2  sides 
proportional,  incl.  A  —  (287),  3  sides  proportional  (288)  ;  in  rt.  A  alti- 
tude forms  A  ~  each  other  and  to  the  whole  (296)  ;  ~  polygons  can 
be  divided  into  ~  A  (293)  ;  polygons  composed  of  ~  A  are  ~  (294). 

V.  AREA    RATIOS.      A  having    an    —    Z   (290) ;     [U  having  an 
=  Z  (290)  ;  ~  A  proportional  to  squares  of  sides  (291)  ;  ~  polygons 
proportional  to  squares  of  sides  (295) ;  equivalent  when  an  angle  — , 
product  including  sides  the  same  (292) . 

CONSTRUCTIONS 

VI.  DIVIDE  A  SECT.     Internally  and  ext.  in  a  given  ratio  (301)  ; 
into  parts  proportional  to  given  sects  (302)  ;  in  mean  and  extreme 
ratio  (307). 

VII.  FIND  A  PROPORTIONAL.     Fourth  to  three  given  sects  (303); 
third  to  two  given  sects  (304)  ;  mean  between  two  given  sects  (305). 

VIII.  CONSTRUCT  A  POLYGON.     Similar  to  a  given  polygon  (308). 

IX.  CONSTRUCT    A   SQUARE.     Having  a  given  ratio  to  a  given 
square  (309). 

311.  ORAL  AND  REVIEW  QUESTIONS 

What  are  the  two  principal  ways  of  getting  sects  proportional? 
Upon  what  do  area  ratios  always  depend?  How  many  parts  are 
needed  to  prove  triangles  similar?  Will  any  three  parts  do?  If  one 
side  of  a  polygon  is  double  the  corresponding  side  of  a  similar  poly- 
gon, what  can  be  told  about  the  areas  ?  If  the  area  is  double,  what 
can  be  told  about  the  side?  If  each  side  of  a  triangle  including  one 
of  the  angles  is  doubled,  what  does  it  do  to  the  area  of  the  triangle? 


156  PROPORTIONS 

if  one  side  is  made  three  halves  as  long,  the  other  two  thirds  as  long? 
How  could  a  triangle  be  made  isosceles  without  changing  the  vertex 
angle  or  the  area?  Why  are  polygons  similar  to  the  same  polygon 
similar  to  each  other?  In  proving  polygons  similar,  what  two  things 
must  be  considered ?  Why  are  regular  polygons  of  the  same  number 
of  sides  mutually  equiangular?  Why  are  their  sides  proportional? 
What  is  used  when  the  ratio  of  a  sum  of  several  magnitudes  to  a 
second  sum  is  needed?  What  changes  can  be  made  in  a  proportion 
without  making  it  untrue?  If  the  sides  of  a  right  triangle  are 
3,  4,  5,  how  long  is  the  projection  of  3  on  5?  of  4  on  5?  How  long 
is  the  altitude  to  5?  How  long  are  the  sects  of  5  made  by  the 
bisector  of  the  opposite  angle  ?  Tell  three  ways  to  construct  a  mean 
proportional.  What  is  the  foundation  of  the  constructions  which 
make  four  sects  proportional  ?  Name  three  such  constructions.  What 
constructions  in  this  book  have  special  cases  which  have  been  done 
before?  What  new  way  of  finding  that  a  line  is  parallel  to  another 
line  occurs  in  this  book?  What  special  case  of  it  has  been  taken 
before?  What  other  theorem  is  the  general  case  corresponding  to 
a  special  case  done  before?  Define  mean  and  extreme  ratio,  cutting 
harmonically,  similar  polygons,  fourth  proportional,  composition, 
division.  In  what  form  is  the  mean  and  extreme  ratio  proportion 
put  before  trying  to  find  the  internal  point  of  division?  the  external? 
What  is  meant  by  a  point  cutting  a  sect  in  a  given  ratio  internally? 
externally?  In  what  ratio  is  it  not  possible  to  cut  a  sect?  What  is 
the  fundamental  way  of  proving  four  sects  proportional?  What  two 
cases  of  it  are  there  ?  If  a  line  were  drawn  through  the  two  thirds 
points  of  the  legs  of  a  trapezoid,  would  it  be  parallel  to  the  bases? 
Why?  What  new  way  of  proving  triangles  equivalent  has  been 
found?  Give  a  numerical  example  illustrating  it.  How  can  a  line 
of  length  \/2  be  constructed?  the  length  Vn?  Is  the  length  exact 
or  approximate  (in  so  far  as  the  instruments  are  accurate)  ? 

GENERAL  EXERCISES 

465.  If  two  lines  through  the  vertex  of  a  tria-ngle  divide  the  op- 
posite side  harmonically,  and  are  at  right  angles  to  each  other,  they 
bisect  the  angles  at  that  vertex. 

466.  Find  the  locus  of  a  point  whose  distances  from  two  given 
points  are  in  a  given  ratio. 


GENERAL  EXERCISES  157 

467.  Find  the  locus  of  a  point  such  that  the  angle  between  the 
tangents  from  the  point  to  one  given  circle  equals  the  angle  between 
the  tangents  from  the  point  to  a  second  given  circle. 

468.  Construct  a  square  inscribed  in  a  semicircle. 

469.  A  line  is  drawn  from  a  vertex  of  a  parallelogram,  cutting  the 
other  diagonal,  and  the  other  two  sides,  one  extended.     Prove  that 
the  sect  from  the  vertex  to  the  diagonal  is  the  mean  between  the  sects 
from  the  diagonal  to  the  sides. 

470.  If  the  center  line  of  two  circles  that  do  not  meet  intersects 
the  exterior  common  tangents  at  P,  and  the  circles  at  A,  B,  C,  D, 
and  a  secant  from  P  meets  the  circles  at  A',  Y,  Z,  W,  then  PX  x  PW 
=  PYx  PZ  =  PB  x  PC. 

471.  Find  the  locus  of  the  vertex  of  a  triangle,  given  the  base,  and 
the  ratio  of  the  other  sides. 

472.  Find  the  locus  of  a  point  whose  distances  from  two  inter- 
secting lines  have  a  given  ratio. 

473.  Find  the  locus  of  a  point  whose  distances  from  the  sides  of  a 
triangle  are  in  a  given  ratio. 

474-   Cut  off  equal  parts  from  two  given  sects,  leaving  them  in  a 
given  ratio. 

475.  Inscribe   a   triangle  similar  to  a  given  triangle  in  a  given 
circle. 

476.  Circumscribe  a  triangle  similar  to  a  given  triangle  about  a 
given  circle. 

477.  From  a  given  point  on   a  side  of  a  triangle,  draw   a  line 
dividing  the  triangle  into  equivalent  parts. 

478.  Construct    an    equilateral    triangle    equivalent  to   a    given 
triangle. 

479.  Construct  a  sect,  given  the  greater  sect  obtained  by  dividing 
it  in  mean  and  extreme  ratio. 

480.  Inscribe  a  square  in  a  given  triangle. 

481.  Inscribe  a  rectangle  similar  to  a  given  rectangle  in  a  given 
semicircle. 

482.  Inscribe  a  rectangle  similar  to  a  given  rectangle  in  a  given 
triangle. 


158  PROPORTIONS 

483.  In  a  triangle  of  sides  a,  b,  c,  find  the  radius  of  the  circum- 
scribed circle. 

484'    Draw  a  square  equivalent  to  a  given  rhombus. 

485.  Describe  a  circle  through  a  given  point,  tangent  to  two  given 
lines. 

486.  Construct  a  triangle,  given  the  three  altitudes.      Note  that 
the  ratio  of  the  altitudes  can  be  used  to  find  the  ratio  of  the  sides. 

487.  Construct  an  isosceles  triangle,  given  the  vertex  angle  and 
the  sum  of  the  base  and  its  altitude. 

488.  Construct  a  triangle,  given  the  base  angles,  and  the  difference 
between  the  base  and  its  altitude. 

NOTE.  In  many  triangle  and  other  constructions,  a  similar  figure 
can  be  used  to  advantage;  as  here,  a  triangle  similar  to  the  required 
triangle  should  be  constructed  first. 

489.  Construct  a  triangle,  given  the  base,  a  base  angle,  and  the 
ratio  of  the  other  including  side  to  the  radius  of  the  circumscribed 
circle. 

490.  Construct  a  triangle,  given  the  three  medians. 

NOTE.  Many  other  triangle  constructions  can  be  solved  by  the 
use  of  proportions  and  of  similar  figures.  It  is  good  practice  for 
students  to  try  to  invent  new  combinations  of  parts  of  triangles  from 
which  the  triangles  can  be  constructed.  See  also  note  at  end  of 
Bk.  II. 


BOOK   V.     REGULAR    POLYGONS    AND 
CIRCLES 

SECTION   I.     THEOREMS 

312.  Theorem  I.     In  any  regular  polygon  there  is  a 
point  that   is  equidistant  from  the  vertices  and  equi- 
distant from  the  sides. 

Prove  that  the  bisectors  of  all  the  angles  meet  in  a 
point,  and  use  locus. 

313.  Center,  Radius,  Apothem.     That  point  in  a  regular 
polygon  which  is  equidistant  from  the  vertices,  and  also 
equidistant  from  the  sides,  is  called  the  center  of  the  poly- 
gon ;  the  line  from  the  center  to  a  vertex  is  called  the 
radius  of  the  polygon,  and  a  perpendicular  from  the  center 
to  a  side  is  called  the  apothem  of  the  polygon. 

It  is  evident  that  the  radius  of  the  polygon  is  also  the 
radius  of  the  circumscribed  circle,  and  that  the  apothem 
of  the  polygon  is  the  radius  of  the  inscribed  circle. 

314.  COR.  1.    Tlw  area  of  a  regular  poly gon  equals  one 
half  tlw  product  of  the  perimeter  by  tlw  apothem. 

315.  Con.  2.    The  perimeters  of  regular  polygons  of  the 
same  number  of  sides  are  proportional  to  their  sides, 
apothems,  or  radii. 

316.  COR.  3.    Tlw  areas  of  regular  polygons  of  the 
same  number  of  sides  are  proportional  to  the  squares  of 
their  sides,  apothems,  or  radii. 

159 


160  REGULAR  POLYGONS   AND   CIRCLES 

Note  again  that  the  area  ratio  is  the  square  of  the  line 
ratio. 

49 1.  If  from  a  point  within  a  polygon  of  n  sides  perpendiculars 
are  drawn  to  all  the  sides,  the  sum  of  those  perpendiculars  is  n  times 
the  apothem. 

492.  Find  the  area  of  a  regular  hexagon  of  side  10  ;  of  side  a. 

493.  If  the  apothem  of  one  regular  polygon  is  12,  that  of  another 
of  the  same  number  of  sides  is  15,  and  the  area  of  the  first  is  477.16, 
find  the  area  of  the  second. 

317.  Theorem  II.     An  equilateral  polygon  inscribed  in 
a  circle  is  regular;   an  equiangular  polygon  circum- 
scribed about  a  circle  is  regular. 

318.  Variables  and  Limits.     A  constant  quantity  is  one 
that  keeps  the  same  value  throughout  the  investigation 
in  question.     A  quantity  may  be  constant  in  'one  discus- 
sion, but  not  in  another. 

A  variable  quantity  is  one  that  takes  different  suc- 
cessive values  during  an  investigation. 

The  limit  of  a  variable  is  that  constant  to  which  the 
variable  can  approach  so  near  that  the  difference  is  less 
than  any  possible  fixed  quantity,  but  which  the  variable 
cannot  equal. 

If  the  sum  of  the  numbers  1,  J,  J,  J,  T^,  -fa,  etc.,  is 
taken,  that  sum  will  never  equal  2,  no  matter  how  large 
a  number  of  terms  is  added.  However,  it  is  not  possible 
to  name  a  number  less  than  2,  such  that  the  sum  cannot 
become  greater  than  that  number ;  that  is,  be  nearer  to  2 
(its  limit)  than  any  fixed  number. 

It  is  evident  from  the  definition  that  when  a  variable 
approaches  its  limit,  the  difference  between  the  limit  and 
the  variable  approaches  the  limit  zero  ;  and,  conversely, 
that  when  the  difference  between  a  constant  and  a  vari- 


THEOREMS  161 

able   approaches   zero   as   a   limit,  the  variable  must    be 
approaching  the  constant  as  its  limit. 

319.  Limit  Theorems.     (Given  without  proof  ;  see  Ap- 
pendix, §  347.) 

(1)  If  two  variables  approaching  limits  are  equal  for 
all  values,  their  limits  are  equal. 

(2)  If  a  variable  is  approaching  a  limit,  that  variable 
multiplied  by,  or  divided  by,  any  constant  will  approach 
its  limit  multiplied  by,  or  divided  by,  that  constant. 

(3)  If  two  variables  are  proportional  to  two  constants, 
their  limits  are  proportional  to  the  same  constants. 

320.  Theorem  III.     If  the  number  of  sides  of  a  regular 
polygon  inscribed  in  a  circle  is  increased  indefinitely,  the 
apothem  of  the  polygon  will  approach  the  radius  of  the 
circle  as  a  limit. 

Show   that  r  —  a  <  — ,   where  r,  a,  s,  stand   for  radius, 

£t 

apothem,    and    side ;   then   show   that   &  =  0,    when    the 
number  of  sides  is  increased  indefinitely. 

321.  Circumference  Axiom.     The   circumference    of   a 
circle  is  the  limit  which  the  perimeters  of  regular  in- 
scribed and  circumscribed  polygons  approach  when  the 
number  of  sides  is  increased  indefinitely. 

322.  Theorem  IV.     The  area  of  a  circle  is  the  limit 
which  the  areas  of  regular  inscribed  and  circumscribed 
polygons  approach  when  the  number  of  sides  is  increased 
indefinitely. 

323.  Theorem  V.     The  ratio  of  the  circumference  to  the 
diameter  is  the  same  for  all  circles  (or,  circumferences 
are  proportional  to  their  diameters'). 

SMITH'S  SYL.  PL.  GEOM. — 11 


162  REGULAR   POLYGONS  AND   CIRCLES 

Regular  inscribed  polygons  have  perimeters  propor- 
tional to  the  diameters  ;  apply  limit  Th.  III. 

324.  Value  of  the  Ratio  of  Circumference  to  Diameter. 

The  ratio  of  circumference  to  diameter  is  represented  by 
the  Greek  letter  TT  (called  pi).  This  letter  is  the  initial 
letter  of  the  Greek  word  for  circumference.  The  value  of 
TT  can  be  found  numerically  by  Geometry,  and  the  method 
employed  is  shown  in  the  Appendix  (§  350);  the  value 
commonly  used  in  calculations  is  3.14159,  or  for  less 
accurate  results,  3.1416,  or  even  3^.  The  last  value  is 
sufficiently  accurate  for  many  of  the  numerical  exercises 
of  the  Geometry,  but  the  student  should  become  accus- 
tomed to  the  use  of  the  more  accurate  values  also. 

The  value  of  TT  cannot  be  expressed  exactly  in  the  deci- 
mal system ;  that  is,  it  is  an  incommensurable  number. 
It  has,  however,  been  calculated  to  over  700  decimal 
places,  and  there  is  no  limit  to  the  accuracy  with  which  a 
calculation  can  be  carried  out,  if  it  is  considered  worth 
while. 

325.  COR.  I.     The  circumference  of  any  circle  equals 

2  TT  tunes  its  radius. 

494-  If  the  radius  of  one  circle  is  double  the  radius  of  a  second 
circle,  and  the  circumference  of  the  second  circle  is  30  ft.  long,  how 
long  is  the  circumference  of  the  first  circle  ? 

495.  What  is  the  width  of  the  ring  between  two  concentric  circles 
whose  circumferences  are  100  and  200  ft. 

496.  If  one  third  of   the'  circumference  of  one  circle  equals  one 
fourth  the  circumference  of  a  second  circle,  how  do  the  radii  compare  V 

497.  Find  the  circumference  of  a  circle  of  radius  10  in. 

498.  Find  the  radius  of  a  circle  of  circumference  22  ft. 

326.  Theorem  VI.     The  area  of  a  circle  is  one  half  the 
product  of  the  circumference  by  the  radius. 

Use  a  circumscribed  regular  polygon. 


THEOREMS  163 

327.  COR.   1.     The  area  of  a  circle  equals  TT  times  the 
square  of  the  radius. 

328.  COR.   2.     The   areas  of  two  circles  are  propor- 
tional to  the  squares  of  their  radii. 

499.  Find  the  area  of  a  circle  of  radius  10. 

500.  Find  the  radius  of  a  circle  of  area  49. 

501.  Find  the  circumference  of  a  circle  of  area  3.14159. 

502.  If  the  radius  of  one  circle  is  four  times  the  radius  of  a  second 
circle,  the  area  of  the  first  is  how  many  times  the  area  of  the  second? 

503.  If  the  circumference  of  one  circle  is  twice  the  circumference 
of  a  second,  how  do  the  areas  compare  ? 

504-   What  is  the  area  of  the  ring  between  concentric  circles  of 
circumferences  100  and  200  ft.? 


SECTION  II.     CONSTRUCTIONS 

329.  CONST.   I.      (1)   To  inscribe  a  circle  in  a  given  reg- 
ular polygon. 

(2)  To  circumscribe  a  circle  about  a  given  regular 
polygon. 

330.  CONST.  II.     (1)    G-iven  a  regular  inscribed  poly- 
gon, to  draw  the  regular  circumscribed  polygon  of  the  same 
number  of  sides. 

(2)  G-iven  a  regular  circumscribed  polygon,  to  draw  the 
regular  inscribed  polygon  of  the  same  number  of  sides. 

331.  CONST.    III.     (1)    G-iven  a  regular  inscribed  poly- 
gon, to  draiv  the  regular  inscribed  polygon  of  double  the 
number  of  sides. 

(2)  G-iven  a  regular  circumscribed  polygon,  to  draw 
the  regular  circumscribed  polygon  of  double  the  number 
of  sides. 

332.  CONST.   IV.      To  inscribe  a  square  in  a  given  circle. 

333.  CONST.     V.      To    inscribe  a  regular  hexagon  in  a 
given  circle. 

334.  CONST.  VI.      To  inscribe  a  regular  decagon  in  a 
given  circle. 

505.   Construct  a  regular  pentagon  inscribed  in  a  given  circle. 

164 


SUMMARY  OF   PROPOSITIONS  165 

506.  Construct  a  regular  12-sided  figure  inscribed  in  a  given  circle. 

507.  Circumscribe  about  a  given  circle  a  polygon  of  4,  5,  6,  8,  10, 
sides. 

335.  Regular  Polygons  which  can  be  Constructed.     It  is 
clear  from  the  constructions  in  this  book  that  polygons  of 
3,  4,  5,  sides,  or  those  numbers  multiplied  by  any  power 
of  2,  are  possible.     There  are  certain  other  regular  poly- 
gons which  can  be   constructed  with  compass  and  ruler, 
such  as  the  polygon  of  17  sides,  but  the  ones  mentioned, 
and  possibly  the  one  of   15  sides,  are  the  only  ones  of 
importance  to  Elementary  Geometry. 

508.  Construct  a  regular  polygon  of  15  sides. 

336.  SUMMARY  OF  THEOREMS  AND  COROLLARIES.    BOOK  V 

(Numbers  in  parentheses  refer  to  black-faced  section  numbers.) 

I.  REGULAR  POLYGON.     Has  a  center  (312);   area  =  \  ap.  x  per. 
(314) ;  perimeters  prop,  to  sides,  apothems,  radii  (315)  ;  areas  prop, 
to  squares  of  sides,  apothems,  radii  (316) ;   equilateral  insc.  polygon 
regular,  equiangular  circumscribed  polygon  regular  (317). 

II.  LIMIT  APPLICATIONS.     Apothem  of  regular  inscribed  polygon 
approaches  radius  (320)  ;  circumference  axiom  (321)  ;   area  regular 
insc.  or  circum.  polygon  approaches  the  area  of  the  circle  (322). 

III.  CIRCLES.     Ratio  of  circumference  to  diameter  (TT)  the  same 
for   all  circles   (323)  ;     circum.  =  2  irr  (325)  ;     area  =  £  r  x  circum. 
(326)  ;  area  —  -rrr2  (327)  ;  areas  prop,  to  squares  of  radii  (328). 

CONSTRUCTIONS 

IV.  To  DRAW  A  CIRCLE.     Inscribed  in  a  given  regular  polygon 
(329)  ;  circumscribed  about  a  given  regular  polygon  (329). 

V.  To   DRAW  A  REGULAR   POLYGON.     Circumscribed,   of    same 
number  of  sides  as  a  given  inscribed,  or  inscribed,  of  the  same  number 
of  sides  as  a  given  circumscribed  (330)  ;  of  double  the  number  of  sides 
of  a  given  polygon,  inscribed  or  circumscribed  (331);  of  4  sides  in- 
scribed (332)  ;  of  6  sides  inscribed  (333);  of  10  sides  inscribed  (334). 


166  REGULAR   POLYGONS   AND   CIRCLES 

337.  ORAL  AND  REVIEW  QUESTIONS 

How  is  the  center  of  a  regular  polygon  found  ?  Tell  a  second  way. 
To  what  are  the  perimeters  of  regular  polygons  proportional?  the 
areas?  the  circumferences  of  circles?  the  areas  of  circles?  Give  a 
formula  for  the  circumference  of  a  circle,  two  formulas  for  the  area 
of  a  circle.  To  what  is  the  side  of  a  regular  inscribed  hexagon  equal? 
the  side  of  a  regular  inscribed  decagon  ?  the  side  of  a  circumscribed 
square?  of  an  inscribed  square?  of  a  circumscribed  equilateral  trian- 
gle? of  an  inscribed  equilateral  triangle?  What  must  be  known  to 
prove  an  inscribed  polygon  regular?  a  circumscribed  polygon?  Ex- 
plain the  inscribed  case.  What  is  the  construction  upon  which  the 
regular  decagon  depends?  How  can  a  regular  pentagon  be  drawn? 
Tell  which  of  the  following  regular  polygons  can  be  constructed  with 
compass  and  ruler,  and  explain;  7,  8,  9, 12, 14, 16  sides.  What  values 
of  TT  are  most  commonly  used?  What  is  the  circumference  of  a  circle 
of  radius  10?  the  area?  How  can  you  find  the  radius  from  the  cir- 
cumference? the  radius  from  the  area?  the  circumference  from  the 
area?  the  area  from  the  circumference?  How  can  circumferences  be 
added?  subtracted?  multiplied?  How  can  the  areas  of  circles  be 
added  ?  subtracted  ?  multiplied  ?  divided  ?  Explain  how  to  draw  a 
circle  having  its  circumference  double  that  of  a  given  circle;  its  area 
double  that  of  the  given  circle. 

GENERAL  EXERCISES 

509.  The  area  of  a  regular  dodecagon  (12)  equals  three  squares 
on  its  radius. 

510.  What  is  the  radius  of  a  circle,  if  the  area  of  the  regular  in- 
scribed hexagon  is  6  V3  ? 

511.  The  area  of  the  ring  between  two  concentric  circles  equals 
that  of  a  circle  whose  diameter  is  that  chord  of  the  larger  which  is 
tangent  to  the  smaller. 

512.  If  two  chords  of  a  circle  are  perpendicular  to  each  other,  the 
sum  of  the  circles  on  the  sects  as  diameters  equals  the  original  circle. 

613.  On  the  sides  of  a  square  of  side  o,  as  diameters,  circles  are 
drawn.     Find  the  area  of  the  parts  into  which  the  square  is  divided. 

614.  On  the  sides  of  an  equilateral  triangle  of  side  a  as  diameters, 
circles  are  drawn.     Find  the  areas  of  the  parts  of  the  figure  formed. 


GENERAL   EXERCISES  167 

51 5.  With  the  vertices  of  an  equilateral  triangle  of  side  a  as  cen- 
ters, and  a  radius  equal  to  half  the  side,  circles  are  drawn.     Find  the 
area  of  the  entire  figure,  and  of  the  figure  inside  the  triangle  bounded 
by  the  arcs. 

516.  Construct  a  regular  octagon  on  a  given  sect  as  side. 
517'   Construct  a  regular  hexagon  on  a  given  side. 

518.  Construct  a  regular  decagon  on  a  given  side. 

51 9.  Construct  a  regular  pentagon  on  a  given  side. 

520.  Construct  a  regular  hexagon,  given  the  shorter  diagonal. 

521.  Construct  a  circle  equal  to  the  sum  of  two  given  circles. 

522.  Construct  a  circle  equal  to  the  difference  of  two  given  circles. 

523.  Construct  a  circle  equal  to  the  sum  of  any  number  of  given 
circles. 

524-    Construct  a  circle  equal  to  any  number  of  times  a  given  circle. 

525.  Construct  a  circumference  equal  to  the  sum  of  two  given  cir- 
cumferences. 

526.  Construct  a  circumference  equal  to  the  difference  of  two  given 
circumferences. 

527.  Construct  a  circumference  equal  to  the  sum  of  any  number  of 
given  circumferences. 

528.  Construct  a  circle  whose  area  has  any  given  ratio  to  the  area 
of  a  given  circle. 

529.  Construct  a  circle  whose  area  is  one  half  the  area  of  a  given 
circle. 

530.  Divide  a  given  circumference  into  parts  having  the  ratio  3  : 7, 
by  a  line  through  a  given  point. 

531.  Divide  the  surface  of  a  circle  into  equal  parts,  by  a  concen- 
tric circle. 

532.  Divide  the  surface  of  a  circle  into  any  number  of  equal  parts 
by  concentric  circles. 

533.  Given  the  radius  of  a  circle,  find  the  perimeters  of  regular 
inscribed  and  circumscribed  polygons  of  3,  4,  5,  6,  8,  10  sides. 

53^.   Cut  off  two  thirds  of  a  circle  by  a  line  through  a  given  point, 


GENERAL 

THE   FORMULAS   OF   GEOMETRY 

338.    I.  IN  AN  n-siDED  POLYGON. 

(1)  The  sum  of  the  interior  angles  equals  (n  —  2)  st.  ^. 

(2)  The  sum  of  the  exterior  angles  equals  2  st.  A. 

II.   ANGLES  FORMED   BY  LINES   MEETING    A   CIRCUMFER- 
ENCE. 

(1)  Vertex  on  the  circumference,  measured  by  half  the 
arc  ;   includes  inscribed  angles  (three  cases),  tangent 
and  chord  angles. 

(2)  Vertex  inside  the  circle,  measured  by  half  the  sum 
of  the  arcs. 

(3)  Vertex  outside  the  circle,  measured  by  half  the  dif- 
ference of  the  arcs;  includes   angles   between  two 
secants,  two  tangents,  tangent  and  secant ;  angle  be- 
tween two  tangents   supplemental  to  central  angle 
between  radii  to  points  of  tangency. 

III.   AREA. 

'(1)  Rectangle,  equals  base  times  altitude. 

(2)  Parallelogram,  equals  base  times  altitude. 

(3)  Triangle,  equals  half  base  times  altitude. 

(4)  Triangle,  given  the  sides,  equals  the  square  root  of 
s(s  —  a)(s  —  b)(s  —  c),  where  s  is  half  the   sum  of 
the  sides. 

(5)  Equilateral  triangle,  equals  the  square  of  half  the 
side  times  the  square  root  of  three. 

(6)  Trapezoid,  equals  half  the  sum  of  the  bases  times 
the  altitude. 

(7)  Regular  polygon,  equals  half  the  perimeter  times 
the  apothem. 

(8)  Circle,  equals  TT  times  the  square  of  the  radius. 

168 


FORMULAS  OF  GEOMETRY         169 

IV.   EQUIVALENCE  FORMULAS  BASED  ON  THE  AXIOM  OF  THE 
WHOLE. 

(1)  The  square  on  the  sura  of  two  sects  is  equivalent  to 
the  sum  of  their  squares  plus  twice  their  rectangle. 

(2)  The  square  on  the  difference  of  two  sects  is  equivalent 
to  the  sum  of  their  squares  less  twice  their  triangle. 

(3)  The  square  on  twice   a  sect   is  equivalent  to  four 
squares  on  the  sect. 

(4)  the  difference  of  the  squares  on  two  sects  is  equiva- 
lent to  the  rectangle  of  their  sum  and  their  difference. 

V.   THE  SQUARE  ON  A  SIDE  OF  A  TRIANGLE. 

(1)  Opposite  a  right  angle,  is  equivalent  to  the  sum  of  the 
squares  on  the  other  sides. 

(2)  Opposite  an  obtuse  angle,  is  equivalent  to  the  sum  of 
the  squares  on  the  other  sides  plus  twice  the  rectangle 
of  one  side  by  the  projection  of  the  other  on  its  line. 

.(3)  Opposite  an  acute  angle,  is  equivalent  to  the  sum  of 
the  squares  on  the  other  sides,  less  twice  the  rectangle 
of  one  side  by  the  projection  of  the  other  side  on  its 
line ;  or,  in  a  right  triangle,  to  difference  of  squares. 

VI.   THE  SUM  OF  THE  SQUARES  ON  Two  SIDES  OF  A  TRIANGLE. 

Is  equivalent  to  twice  the  sum  of  the  squares  on  one  half 
the  third  side  and  on  the  median  to  that  side ;  difference, 
to  two  rectangles  of  the  base  by  its  median's  projection. 

VII.   PROPORTIONS  BETWEEN  SECTS. 

(1)  Parallels  cutting  transversals,  including  triangle  case. 

(2)  Sides  of  similar  polygons  are  proportional. 

(3)  In  a  right  triangle  with  the  altitude  to  the  hypotenuse, 
(a)  the  altitude  is  the  mean  between  the  sects  of  the 

hypotenuse  ; 

(ft)  either  leg  is  the  mean  between  the  hypotenuse  and 
the  projection  of  that  leg  on  the  hypotenuse. 

(4)  If  two  secants  cut  a  circle,  the  product  of  the  sects  from 
the  vertex  is  the  same  ;  a  tangent  is  the  mean  between 
the  sects  of  a  secant  from  the  same  point. 

(5)  The  bisector  of  an  angle  of  a  triangle  cuts  the  opposite 
side  (internally  or  externally)  into  sects  proportional  to 
the  sides  including  the  angle. 


170          FORMULAS  OF  GEOMETRY 

(6)  Perimeters  of  regular  polygons  are  proportional  to 
sides,  apothems,  or  radii. 

(7)  Circumferences  are  proportional  to  radii. 

VIII.   PROPORTIONS  BETWEEN  AREAS. 

(1)  Areas  of  triangles  having  an  angle  equal  are  propor- 
tional to  the  product  of  the  sides  including  the  angle ; 
if  the  triangles  are  equivalent,  the  product  of  the  sides 
in  one  equals  the  product  in  the  other,  and  conversely. 

(2)  Areas  of  any  similar  polygons  are  proportional  to  the 
squares  of  corresponding  sides. 

(3)  Areas  of  circles  are  proportional  to  the  squares  of  their 
radii. 

IX.   THE  CIRCUMFERENCE  OF  A  CIRCLE  equals  2  irR. 
X.   ALTITUDE. 

(1)  Of  any  triangle,  is  found  from  III,  4,  or  by  first  find- 
ing the  projection  in  V,  (2)  or  (3). 

(2)  In  an  equilateral  triangle,  equals  half  the  side  times 
the  square  root  of  three. 

XL   PROJECTION  OF  A  SIDE  OF  A  TRIANGLE,  from  V,  (2),  or  (3). 
XII.  MEDIAN  OF  A  TRIANGLE,  from  VI. 


COLLEGE   EXAMINATION   QUESTIONS 

535.  If  ABODE  is  an  inscribed  pentagon,  and  arc  DE  is  40°, 
find  the  number  of  degrees  in  the  sum  of  angles  A  and  C. 

536.  In  the  triangle  ARC,  AB  equals  7,  BC  equals  8,  and  CA 
equals  5.    Find  the  projection  of  BC  on  CA. 

537.  Construct  a  square  whose  area  is  to  that  of  a  given  square  as 
2  is  to  5. 

538.  Construct  a  circumference  equal  to   the  difference  of  two 
given  circumferences. 

539.  Construct  a  triangle,  given  two  sides  and  the  altitude  on  one 
of  those  sides. 

540.  In  a  circle  of  center  O,  two  perpendiculars,  OD  and  OE,  are 
drawn  to  chords  AB  and  CF,  respectively,  and  angle  EDO  equals 
angle  DEO.     Prove  AB  equal  to  CF. 

541.  Upon  the  four  sides  of  a  square   as  chords  arcs  of  90°  are 
constructed  within  the  square.     If  the  diagonal  of  the  square  equals 
2  M  in.,  find  (a)  the  area  bounded  by  the  four  arcs ;  (b)  the  radius 
of  a  circle  equal  to  this  area. 

542.  Two  equivalent  triangles  have  a  common  vertex  y,  and  equal 
bases  AB  and  CD.     If  A,  B,  C,  D  have  fixed  positions  not  in  the 
same  Hue,  find  the  locus  of  the  point  Y. 

543.  The  side  of  an  inscribed  equilateral  triangle  is  parallel  to  the 
side  of  a  regular  decagon  inscribed  in  the  same  circle.      Find  the 
number  of  degrees  contained  in  the  intercepted  arcs,  if  the  center  of 
the  circle  lies  between  the  two  lines. 

544.  If  the  sides  of  a  triangle  are  7  and  9  cm.,  respectively,  and  the 
median  to  the  third  side  equals  7  cm.,  find  the  third  side. 

545.  The  sides  of  a  polygon  are  4,  5,  6,  7,  8  cm.     Find  the  sides 
of  a  similar  polygon  whose  area  equals  four  times  the  area  of  the 
given  polygon. 

171 


172  EXAMINATION   QUESTIONS 

546.  Two  vertices  A   and  B  of  a  triangle  have  fixed  positions. 
Find  the  locus  of  the  third  vertex  C,  if  angle  C  is  equal  to  a  given 
angle  M,  and  prove  your  result. 

547.  If  the  base  of  an  isosceles  triangle  be  trisected,  and  from  each 
point  of  division  a  perpendicular  be  drawn  upon  the  nearest  side, 
prove  the  perpendiculars  equal.     State  the  converse. 

548'  State  two  propositions  which  may  be  used  to  prove  the 
equality  of  arcs.  If  the  lines  which  join  a  point  in  a  circumference 
to  the  midpoints  of  two  radii  are  equal,  prove  that  the  point  bisects 
the  arc  subtended  by  the  two  radii. 

549.  Describe  briefly  a  method  for  proving  that  the  product  of 
two  lines  is  equal  to  the  product  of  two  other  lines.     If  through  a 
point  A  in  a  circumference,  a  tangent  AB  and  a  chord  AC  be  drawn, 
and  from  C  a  diameter  CD  and  a  perpendicular  CE  upon  AB  be 
drawn,  prove  that  the  square  of  A  C  equals  CE  times  CD. 

550.  Construct  a  circle  whose  area  equals  five  times  the  area  of  a 
given  circle. 

551 .  Prove  that  the  square  of  a  line  drawn  from  the  vertex  of  an 
isosceles  triangle  to  any  point  in  the  base  is  equal  to  the  square  of 
the  leg,  diminished  by  the  product  of  the  segments  of  the  base. 

552.  Determine   how  many  sides  the  polygon    has,   the   sum   of 
whose  interior  angles  equals  the  sum  of  its  exterior  angles.     Explain 
the  method  in  full. 

553.  If  a  fixed  arc  AB  of  a  circle  equals  120°,  and  a  movable 
arc  CD  on  arc  BA  equals  60°,  find  the  locus  of  the  intersection  of 
(a)  AC  and  BD;  (b)  AD  and  BC. 

554-  A  chord  24  in.  long  is  9  in.  from  the  center  of  a  circle.  Find 
the  length  of  the  tangents  drawn  from  the  extremities  of  the  chord 
and  produced  till  they  meet. 

555.  In  the  triangle  ABC,  the  angle  A  is  acute  and  BD  is  drawn 
perpendicular  to  AC.     AB  equals  10  ft.,  BC  equals  12  ft.,  and  AC 
equals  14  ft.     Find  A  D. 

556.  The  bases  of  a.  trapezoid  are  8  in.  and  12  in.,  the  area  30 
sq.  in.     Find  the  length  of  a  line  drawn  between  the  legs  parallel  to 
the  lower  base  and  2  in.  from  it. 

557.  Find  the  area  of  the  sector  of  a  circle  whose  radius  is  15 
in.,  the  angle  of  the  sector  being  32°. 


EXAMINATION   QUESTIONS  173 

558.  Show  how  to  construct  a  circle  of  given  radius  tangent  to 
two  given  circles. 

559.  Show  how  to  draw  a  line  parallel  to  the  base  of  an  isosceles 
triangle  so  that  in  the  trapezoid  thus  formed  the  legs  shall  be  equal 
to  the  upper  base. 

560.  Prove  that  the  perpendiculars  dropped  from  the  midpoints 
of  two  sides  of  a  triangle  to  the  third  side  are  equal. 

561.  Two  equal  chords  produced  meet  outside  the  circle.    Prove 
that  the  secants  thus  formed  are  equal. 

562.  Find  the  locus  of  the  point  at  which  a  given  segment  of  a 
straight  line  subtends  a  given  constant  angle. 

563.  A  quadrilateral  is  formed  by  the  diameter  AB  of  a  circle, 
the  tangents  at  .4  and  B,  and  a  third  tangent  which  meets  the  other 
tangents  at  C  and  D,  respectively.    Prove  that  the  area  is  equal  to 
one  half  the  product  of  the  opposite  sides  AB  and  CD. 

564.  A  circle  of  radius  4  has  its  center  at  the  intersection  of  the 
diagonals  of  a  square  whose  side  is  12.    Find  the  length  of  the  circum- 
ference of  a  circle  which  touches  two  adjacent  sides  of  the  square  and 
also  the  circle. 

565.  How  many  sides  has  the  polygon  each  of  whose  interior  angles 
is  175°? 


566.  If  M  and  N  are  two  lines,  construct  a  line  equal  to  V2  MN. 

567.  If  triangle  ABC  is  equivalent  to  triangle  DEF,  A  equal  to 
D,  AB  =  6,  and  DE  =  9,  find  the  ratio  of  CA  to  FD. 

568.  ABC  is  a  right  triangle,  C  the  right  angle ;  EC  =  5,  CA  =  12, 
AB  is  extended  to  D  so  BD  =  10.     If  CB  is  extended  to  E,  and  DE 
drawn,  how  long  must  BE  be  in  order  to  make  triangle  BED  equiva- 
lent to  triangle  ABC. 

569.  If  a  quadrilateral  ABCD  is  inscribed  in  a  circle,  and   the 
diagonals  A  C  and  BD  meet  in  E  so  that  BE  =  CE,  prove  that  arc 
AB  =  arc  CD. 

570.  Construct  a  triangle,  having  given  a  side  and  the  medians  to 
the  other  two  sides. 

571.  Each   side  of   a  triangle  is  2n  cm.,  and   from  each  vertex 
as  a  center  with  radii  equal  to  n  centimeters,  circles  are  drawn.    Find 
the  areas  bounded  (a)  by  the  three  arcs  that  lie  without  the  triangle ; 
(&)  by  the  three  arcs  that  lie  within  the  triangle. 


174  EXAMINATION  QUESTIONS 

572.  Given  a  square  with  the  side  3  in.  long.      Find  the  locus 
of  a  point  P  such  that  the  distance  from  P  to  the  nearest  point  on 
the  perimeter  of  the  square  is  1  in.     Describe  the  locus  accurately. 

573.  Semicircles  are  drawn  with  their  centers  at  the  middle  points 
of  the  sides  of  an  equilateral  triangle,  forming  a  triangle  less  the  area 
of  three  semicircles.     Prove  that  if  the  perimeter  of  this  figure  is  one 
fifth  greater  than  that  of  the  triangle,  its  area  is  about  one  third  less 
than  that  of  the  triangle.      If   the   side  of   the  triangle  is  10  in., 
what  is  the  area  of  the  figure,  correct  to  1  per  cent  ? 

574'  On  the  sides  of  an  equilateral  triangle  ARC  as  bases,  equal 
isosceles  triangles  ABP,  ACQ,  BCR,  are  constructed  ;  the  first  two  are 
exterior  to  the  given  triangle,  while  R  is  on  the  same  side  of  .BC  as 
A.  Prove  that  A  PR  Q  is  a  rhombus. 

575.   Find  the  altitude  of  an   equilateral  triangle  whose  area  is 


576.  If  the  non-parallel  sides  of  a  trapezoid  are  equal,  the  angles 
which  they  make  with  the  base  are  equal. 

577.  Prove  that  the  bisectors  of  the  angles  of  a  quadrilateral  form 
a  second  quadrilateral  of  which  the  opposite  angles  are  supplementary. 

578.  Through  the  point  M  in  the  base  of  a  triangle  parallels  to 
the  other  two  sides  are  drawn,  forming  a  parallelogram.     Find  the 
locus  of  the  center  of  the  parallelogram  as  M  moves  along  the  base  of 
the  triangle. 

579.  Find  to  one  place  of  decimals  the  area  of  a  six-pointed  star 
formed  by  joining  the  alternate  vertices  of  a  regular  hexagon  inscribed 
in  a  circle  of  radius  3  in. 

580.  The  three  angles  of  a  triangle  are  48°,  82°,  and  50°.    Find  the 
three  angles  formed  by  the  bisectors  of  the  angles  of  the  triangle. 
Verify  by  using  the  theorem  involving  the  sum  of  the  angles  about  a 
point  in  a  plane. 

581.  A  chord   1  ft.  long  is  4  in.  from  the   center   of   a  circle. 
How  far  from  the  center  is  a  chord  9  in.  long? 

582.  A  circle  has  an  area  of  80  sq.  ft.     Find  the  length  of  an 
arc  of  80°. 

583.  Prove  that  if  a  median  of  a  triangle  is  equal  to  half  the  side 
to  which  it  is  drawn,  the  triangle  is  a  right  triangle. 


EXAMINATION  QUESTIONS  175 

584.  Prove  that  if  AB  is  a  diameter  of  a  circle,  and  BC  a  tangent, 
and  A  C  meets  the  circumference  at  D,  the  diameter  is  a  mean  pro- 
portional between  AC  and  AD. 

585.  Given  three  lines  a,  6,  c.     Construct  a  line  x  so  that  a  :  b  :  :c  :x. 

586.  Two  parallelograms  are  equal  if  two  sides  and  the  included 
angle  of  one  are  equal  to  two  sides  and  the  included  angle  of  the  other. 

587.  The  area  of  a  regular  inscribed  hexagon  is  a  mean  propor- 
tional between  the  areas  of  the  inscribed  and  circumscribed  equilateral 
triangles. 

588.  Three  equal  circles  are  described  each  tangent  to  the  other 
two.     If  the  common  radius  is  /?,  find  the  area  contained  between 
the  circles. 

589.  The  radius  of  a  circle  is  10  ft. ;  the  area  of  a  sector  of  that 
circle  is  130  sq.  ft.     What  is  its  arc  in  degrees? 

590.  Two  sides  and  a  diagonal  of  a  parallelogram  are  7,  9,  and  8, 
respectively.     Find  the  length  of  the  other  diagonal. 

591.  One  of  two  secants  meeting  without  a  circle  is  12.5  in.,  and 
its  external  segment  is  4  in. ;  the  other  secant  is  divided  into  two 
equal  parts  by  the  circumference.     Find  the  length  of  the  second 
secant. 

592.  A  BCD  is  any  parallelogram,  and   E   any  point  within  it. 
Prove  that  the  sum  of  the  triangles  EAD  and  EBC  equals  half  the 
area  of  the  parallelogram. 

593.  Three  consecutive  angles  of  an  inscribed  quadrilateral  sub- 
tend arcs  of  70°,  85°,  and  98°,  respectively.     Find  each  angle  of  the 
quadrilateral  and  the  angle  between  the  diagonals. 

594-  The  diameters  of  two  concentric  circles  are  16  and  40  ft., 
respectively.  Find  the  length  of  a  chord  of  the  larger  which  is 
tangent  to  the  smaller. 

595.  The  area  of  a  rhombus  is  96  sq.  ft.  and  its  side  is  10  ft. 
Find  the  lengths  of  its  diagonals. 

596.  Show  how  to  inscribe  a  circle  in  a  given  sector. 

597.  Show  how  to  divide  a  triangle  into  two  equivalent  parts  by 
a  line  parallel  to  one  of  its  sides. 

598.  Prove  that  the   tangents  to  two   intersecting  circles  from 
any  point  in  their  common  chord  produced  are  equal. 


176  EXAMINATION   QUESTIONS 

599.  Two  tangents   drawn   from   the   same  external   point  to  a 
circle  form  an  angle  01   64°.     Find  the  number  of  degrees  in  each 
of  the  arcs  intercepted  by  these  tangents. 

600.  Find  the  radius  of  a  circle  whose  circumference  numerically 
equals  its  area. 

601.  From  a  point  12  in.  from  the  center  of  a  circle  16  in.  in 
diameter,  two  tangents  are  drawn  to  the  circumference.     Find  the 
length  of  the  chord  joining  the  points  of  contact. 

602.  Prove  that  if  a  circle   is   circumscribed  about  an  isosceles 
triangle,  the  tangents  through  the  vertices  form  another  isosceles 
triangle. 

603.  Two   circles   are  tangent  externally  and  through  the  point 
of  contact  two  straight  lines  are  drawn  terminating  in  the  circum- 
ferences.    Prove  that  the  corresponding  segments  of   the  lines  are 
proportional. 

604.  Show  how  to  draw  a  line  terminating  in  the  arms  of  an 
angle,  which  shall  be  equal  to  one  given  line  and  parallel  to  another. 

605.  Two  circles  are  tangent  internally.     If  chords  of  the  larger 
circle  are  drawn  through  the  point  of  contact,  prove  that  they  are 
divided  proportionally  by  the  smaller  circle. 

606.  Prove  that  if  one  acute  angle  of  a  triangle  is  double  another, 
the  triangle  can  be  divided  into  two  isosceles  triangles  by  a  straight 
line  through  the  vertex  of  the  third  angle. 

607.  Given  a  regular  hexagon  each  side  of  which  is  6  in.     With 
three  of  the  alternate  vertices  as  centers,  arcs  of  circles  are  drawn 
passing  through  the  center  of  the   polygon.     Find  the  area  of   the 
three  loops  thus  formed. 

608.  A  point  A  is  4  ft.  from  the  circumference  of  a  circle ;  the 
length  of  a  tangent  from  A  to  the  circle  is  10  ft.     Find  the  diameter. 

609.  The  bases  of  a  trapezoid  are  respectively  29  ft.  and  37  ft. 
and  its  area  is  247.5  sq.  ft.     Find  its  altitude. 

610.  Show  how  to  divide  a  given  rectangle  into  four  equivalent 
parts  by  lines  drawn  from  one  of  the  vertices  of  the  rectangle. 

611.  Given  a  straight  line  and  two  points  on  the  same  side  of  that 
line  and  at  unequal  distances  from  it.     Construct  a  circumference 
passing  through  the  two  points  and  having  its  center  in  the  given 
line. 


EXAMINATION   QUESTIONS  177 

612.  Prove  that  the  area  of  a  square  inscribed  in  a  given  circle  is 
twice  the  square  of  the  radius  of  the  circle. 

613.  ABC  is  a  triangle  inscribed  in  a  circle  with  center  0.     Take 
D  the  middle  point  of  the  arc  EC  and  draw  OD  and  AD.     Prove  that 
the  angle  ADO  equals  half  the  difference  of  the  angles  B  and  C. 

614.  If  a  line  is  drawn  from  the  vertex  A  of  triangle  ABC  to  any 
point  D  of  the  opposite  side,  and  any  point  0  on  AD  is  joined  to 

A  ABC      AD 
£ande,  prove  __  =  _. 

615.  From  a  point  at  a  distance  of  10  in.  from  the  center  of  a 
circle  of  radius  5  in.,  two  tangents  are  drawn.     Compute  the  area 
bounded  by  the  tangents  and  their  included  arc. 

616.  Construct  a  triangle  having  given  the  midpoints  of  its  three 
sides. 

617.  The  legs  of  a  right  triangle  are  6  and  8,  respectively.     Find 
their  projections  on  the  hypotenuse. 

618.  Compute  the  area  of  a  square  inscribed  in  a  circle   whose 
perimeter  is  63  ft. 

619.  The  legs  of  a  right  triangle  are  respectively  15  ft.  and  8  ft. 
Find  the  length  of   each  segment  made  on   the  hypotenuse  by  the 
bisector  of  the  right  angle. 

620.  Find   the    product  of    the    segments   of    any  chord   drawn 
through  a  point  12  in.  from  the  center  of   a  circle  whose  diameter 
is  18  in. 

621.  Find  the  area  of  an  equilateral  triangle  inscribed  in  a  circle 
of  radius  4  in. 

622.  Show  how  to  construct  a  square  which   shall  be  one  third 
of  a  given  square. 

623.  Prove  that  if  in  a  right  triangle  one  angle  is  30°,  the  hypote- 
nuse is  double  the  shorter  leg. 

624.  Prove  that  two  chords  drawn  perpendicular  to  a  third  chord 
at  its  extremities  are  equal. 

625.  Draw  a  square  ABCD.     On  the  diagonal  AC  take  the  point 
E  so  that  AE  —  AB,  and  draw  through  E  a  perpendicular  to  AE, 
cutting  BC  in  F.     Prove  that  BF  =  FE  =  EC. 

SMITH'S  SYL.  PL.  GEOM.  — 12 


178  EXAMINATION   QUESTIONS 

626.  Take  five  points  on  the  circumference  of  a  circle  (^4,  5,  C,  Z),  Z?), 
in  the  order  named.     Let  the  middle  point  of  the  arc  ABC  be  F, 
and  the  middle  point  of  the  arc  AED  be  G.     Let  the  chord  FG  cut 
the  chord  A  C  in  H  and  the  chord  AD  in  K.     Prove  that  AH  is  equal 
fc>4& 

627.  Prove  that  if  in  a  right  triangle  the  hypotenuse  is  double 
the  shorter  leg,  one  acute  angle  is  double  the  other. 

628.  Show  how  to  construct  a  square  equivalent  to  a  given  paral- 
lelogram.    Prove  the  correctness  of  your  method. 

629.  Prove  that  if  one  leg  of  a  right  triangle  is  the  diameter  of  a 
circle,  the  tangent  at  the  point  where  the  circumference  cuts  the 
hypotenuse  bisects  the  other  leg. 

630.  The  apothem  of  a  regular  hexagon  is  6\/3.     Find  the  area 
of  the  hexagon  and  the  area  of  the  inscribed  circle. 

631.  Prove  that  if  a  tangent  be  drawn  from  one  end  of  a  diameter 
meeting  a  secant  from  the  other  end,  the  product  of  the  secant  and 
the  internal  segment  will  be  the  same  for  all  directions  of  the  secant. 

632.  Prove  that  if  two  circles  are  tangent  internally  at  A  and  a 
straight  line  intersects  the  two  circles  in  B,  C,  Z>,  and  E,  then  angle 

=  DAE. 


633.  If   in  triangle  ABC  the  line  from   C  to  D  on  AB  bisects 
angle  C,  and  AB  =  2.09,  £C  =  3.14,  CA  =4,  compute  values  of  DB 
and  DA,  respectively. 

634.  To  measure  the  height  of  a  church  spire,  a  rod  10  ft.  long 
is  planted  vertically  at  position  A,  then  at  position  B.     In  each  case 
the  observer  takes  such  a  position  that  the  top  of  the  spire,  the  top 
of  the  rod,  and  his  eye  are  all  in  line  when  he  stands  erect.     He 
measures  the  distance  each  time  from  his  position  to  the  rod.  and 
obtains  the  measurements  4  ft.  and  8.3  ft.     He  also  measures  the 
distance  between  his  two  positions,  finding  it  to  be  138  ft.     If  the 
observer's  eye   is   5  ft.    above  the   ground,   what  is   the   height  of 
the  spire?     Explain  your  solution. 

635.  What  is  the  locus  of   the   midpoint  of  one  leg  of  a  right 
triangle  whose  hypotenuse  is  fixed?     Prove  the  correctness  of  your 
answer. 

636.  In  a  quadrilateral  ABCD  the  lengths  of  AB  and  EC  are 


EXAMINATION   QUESTIONS  179 

equal  and  angle  A   is  greater  than  angle  C.     Which  is  the  longer 
side,  AD  or  CZ>?     Give  the  reason  for  your  answer. 

637.  Prove  that  in  a  convex  quadrilateral  the  angle  between  the 
bisectors  of  two  adjacent  angles  is  one  half  the  sum  of  the  other  two 
angles. 

638.  An  arc  of  a  certain  circle  is  100  ft.  long,  and  subtends  an 
angle  of  25°  at  the  center.     Compute  the  radius  of  the  circle  correct 
to  three  significant  figures. 

639.  Three  successive  vertices  of  a  regular  octagon  are  A,  B,  C, 
respectively.     If  the  length  AB  is  a,  compute  the  length  A  C. 

640.  The  areas  of  similar  segments  of  circles  are  proportional  to 
the  squares  of  their  radii. 

641.  Given  a  circle  whose  radius  is  16.     Find  the  perimeter  and 
the  area  of  the  regular  inscribed  octagon. 

61+2.  Two  circles  intersect  at  points  A  and  B.  Through  A  a 
variable  secant  is  drawn,  cutting  the  circles  in  C  and  D.  Prove  that 
the  angle  CBD  is  constant  for  all  positions  of  the  secant. 

643.  Let  A  and  B  be  two  fixed  points  on  the  circumference  of  a 
given  circle  and  P  and  Q  the  extremities  of  a  variable  diameter  of 
the  same  circle.  Find  the  locus  of  the  point  of  intersection  of  the 
straight  lines  AP  and  BQ. 

644-  Prove  that  in  any  right  triangle  the  line  drawn  from  the 
right  angle  to  the  middle  of  the  hypotenuse  is  equal  to  one  half  the 
hypotenuse. 

645.  The   area  of   a  regular   decagon  is  108  sq.  in.      Find  the 
radius  of  the  circumscribed  circle. 

646.  Two  secants  are  drawn  from  the  same  point  to  the  same 
circle.     The  external  segment  of  the  first  is  5  in.  and  its  internal 
segment  is  19  in.     The  internal  segment  of  the  other  secant  is  7  in. 
Find  the  length  of  the  second  secant. 

647.  On  the  diameter  AB  of  a  circle  mark  a  point  P.     Through 
P  draw  the  chord  CPD  at  right  angles  to  AB.     Prove  that  if  AP, 
BP,  CP,  and  DP  be  taken  as  diameters  of  circles,  the  sum  of  the 
areas  of  the  four  circles  is  equal  to  the  area  of  the  original  circle. 

648.  To  construct  a  rectangle,  having  given  the  perimeter  and  the 
diagonal. 


180  EXAMINATION  QUESTIONS 

649.  Find  how  far  from  the  base  of  a  triangle  of  altitude  a  lines 
parallel  to  the  base  must  be  drawn  to  divide  the  area  of  the  triangle 
into  three  equal  parts. 

650.  Prove  that  two  triangles  are  similar  if  the  sides  of  one  are 
respectively  parallel  to  the  sides  of  the  other. 

651.  Derive  the  numerical  value  of  ?r.. 

652.  Of  all  triangles  having  the  same  base  and  equal  areas,  the 
isosceles  triangle  has  the  minimum  perimeter. 

653.  How  high  is  a  tree  which  casts  a  shadow  70  ft.  long,  when  a 
man  6  ft.  high  casts  a  shadow  8|  ft.  long? 

654.  Construct  a  triangle,   given  the  base,  vertical  angle,   and 
median  drawn  to  the  base. 

655.  State  six  propositions  concerning  parallel  lines  and  prove 
any  one  of  them. 

656.  An  interior  angle  of  an  equiangular  polygon  is  150°.     Find 
the  number  of  sides. 

657.  Three  cylindrical  barrels,  diameter  of  each  being  20  in.,  are 
placed  in  a  pile  with  axes  horizontal  so  that  each  just  touches  the 
other  two.     Find  the   height  of  the  pile,   and   the   length  of  the 
shortest  rope  to  go  over  the  pile  and  touch  the  floor  on  each  side. 

658.  AB  is  the  diameter  of  a  circle  of  radius  2  in.,  and  A  C  is  a 
chord  such  that  BAG  is  30°.     Find  area  and  perimeter  of  BACB 
correct  to  two  decimal  places. 

659.  If  from  a  fixed  point  D,  within  a  triangle  ABC,  lines  are 
drawn  to  all  points  in  the  perimeter  of  the  triangle,  what  is  the  locus 
of  the  middle  points  of  those  lines? 

660.  Show  that  if  the  radius  of  a  circle   is  a,  the  side  of  the 
regular  inscribed  decagon  is    ?(V5—  1)   and  the  side  of  the  regu- 


lar inscribed  pentagon  is  ^ 

661.  The  base  of  a  triangle  is  32  ft.  and  its  height  is  20  ft.    What 
is  the  area  of  the  triangle  formed  by  drawing  a  line  parallel  to  the 
base  5  ft.  from  the  vertex  ? 

662.  The  sides  of  a  triangle  are  5,  12,  13.     Find  the  segments  into 
which  each  side  is  divided  by  the  bisector  of  the  opposite  angle. 


EXAMINATION   QUESTIONS  181 

663.  The  sides  of  a  triangle  are  a,  J,  c,  and  the  area  is  k.     What 
is  the  radius  of  the  inscribed  circle  ? 

664.  A  and  B  are  fixed  points,  AC  is  drawn  in  any  direction,  and 
BP  is  drawn  perpendicular  to  A  C,  meeting  it  at  P.     What  is  the 
locus  of  P  ? 

665.  Upon  a  line  about  ]£  in.  long  construct  a  segment  to  contain 
an  angle  of  60°. 

666.  Three  circles  of  radius  a  touch  each  other,  and  another  circle 
is  circumscribed  about  them.      Find  its  radius,  circumference,  and 
area. 

667.  AB  is  a  fixed  line,  angle  A  CB  =  45°.     Construct  the  locus 
of  C. 

668.  Two  sides  of  a  triangle  are  a  and  b,  the  included  angle  135°. 
What  is  the  square  of  the  third  side  c  ? 

669.  The  lengths  of  the  circumferences  of  two  concentric  circles 
differ  by  6  in.     Compute  the  width  of  the  ring  to  three  significant 
figures. 

670.  Prove  the  area  of  the  triangle  formed  by  joining  the  middle 
point  of  one  of  the  nonparallel  sides  of  a  trapezoid  to  the  extremities 
of  the  opposite  side  equals  one  half  the  trapezoid. 

671 .  The  three  sides  of  a  triangle  are  4  ft.,  13  ft.,  and  15  ft.  long. 
Show  that  the  altitude  upon  the  side  of  length  15  is  3.2. 

672.  Through  the  vertex  A  of  the  parallelogram  A  BCD  draw  a 
secant.     Let  this  line  cut  diagonal  BD  in  E,  and  the  sides  BC,  CD 
(or  these  sides  produced)  in  F  and  G,  respectively.     Prove  that  AE 
is  a  mean  proportional  between  EF  and  EG. 

673.  Let  ABC  be  an  equilateral  triangle,  and  on  the  sides  AB, 
BC,  CA,  lay  off  AD,  BE,  CF,  each  equal  to  one  third  AB,  and  join 
the  points  D,  E,  F,  with  one  another.     Prove  that  the  triangle  DEF 
is  equilateral,  and  that  its  sides  are  respectively  perpendicular  to 
the  sides  of  the  given  triangle. 

674-  On  the  circumference  of  a  circle  take  two  points  subtending 
a  right  angle  at  the  center,  and  a  third  point  on  the  arc  between 
these  two.  Prove  that  the  perimeter  of  the  triangle  formed  by  the 
tangents  at  these  three  points  is  equal  to  the  diameter  of  the  circle. 

675.  An  indefinite  straight  line  moves  in  such  a  way  that  it 
always  passes  through  at  least  one  vertex  of  a  given  square,  but 


182  EXAMINATION   QUESTIONS 

never  crosses  the  square.  What  is  the  locus  of  the  foot  of  the  per- 
pendicular dropped  on  the  moving  line  from  the  center  of  the  square  ? 
Describe  the  locus  accurately  and  prove  the  correctness  of  your' 
answer. 

676.  Prove   the    correctness   of    the    following    construction    for 
bisecting  an   angle  ABC]   upon  AB  produced   beyond  B  take  BD 
equal  to  BC  and  draw  a  line  through  B  parallel  to  DC. 

677.  Show  how  to  construct  a  chord  through  a  given  point  A 
within  a  circle,  so  that  the  extremities  of  the  chord  shall  be  equi- 
distant from  another  point  B. 

678.  A  rod  of  length  a  is  free  to  more  within  a  semicircular  area 
of  radius  a.     Describe  accurately  the  boundary  of  the  region  within 
which  the  middle  point  of  the  rod  will  always  be  found. 

679.  A  roadway  60  ft.  wide  is  cut  through  the  middle  of  a  circu- 
lar field  120  ft.  in  diameter.     Compute  the  area  of  the  remainder  of 
the  field  correct  to  1  per  cent  of  its  value. 

680.  The  radii  of  two  circles  are  1  in.  and  V3  in.,  respectively, 
and  the   distance   between    their   centers  is  2   in.     Compute   their 
common  area  to  three  significant  figures. 

681.  Determine  a  point  P  without  a  given  circle  so  that  the  sum 
of  the  tangents  from  P  to  the  circle  shall  be  equal  to  the  distance 
from  P  to  the  farthest  point  of  the  circle. 

682.  The   image   of  a  point  in  a  mirror  is,  apparently,  as  far 
behind  the  mirror  as  the  point  itself  is  in  front.     If  a  mirror  revolves 
about  a  vertical  axis,  what  will  be  the  locus  of  the  apparent  image 
of  a  fixed  point  one  foot  from  the  axis? 

683.  The  hypotenuse  of  a  right  triangle  is  10  in.  long  and  one  of 
the  acute  angles  is  30°.     Compute  the  lengths  of  the  segments  into 
which  the  short  side  is  divided  by  the  bisector  of  the  opposite  angle. 

684-  A  chord  BC  of  a  given  circle  is  drawn,  and  a  point  A  moves 
on  the  longer  arc  BC.  Draw  triangle  ABC  and  find  the  locus  of  the 
center  of  a  circle  inscribed  in  this  triangle. 

685.  Three  equal  circular  plates  are  so  placed  that  each  touches 
the  other  two,  and  a  string  is  tied  tightly  around  them.  If  the 
length  vof  the  string  is  10  ft.,  find  the  radius  of  the  circles  correct 
to  three  figures. 


APPENDIX 

339.  Contraposite  Law.     This  law  was  stated  in  §  8,  bat 
no  explanation  of  why  it  was  true  was  attempted.     The 
following  explanation  is  simple,  and  shows  very  plainly 
that  the  law  holds  for  all  statements. 

Given.         If  A,  then  B. 

To  prove.    If  not  B,  then  not  A. 

Proof.     I.    If  not  B,  then  either  (1)  A, 

or  (2)  not  A  (all  possibilities). 
II.    But,  if  A,  then  B  (given). 

.-.  using  (1),  if  not  £,  then  A,  then  B. 
III.    This  is  impossible,  for  it  contradicts  itself. 
.-.  If  not  B,  then  not  A 
(only  other  possibility). 

(This  proof  depends  upon  the  "Law  of  Excluded 
Mean,"  but  for  the  purpose  of  this  proof  it  is  not 
necessary  to  discuss  that  Law.) 

340.  Law  of  Converse.     Stated  in  §  10. 
Given.          If  -4,  then  JT, 

If  B,  then  F, 
If  <7,  then  Z,  etc. 

Where  A,  B,  <7,*.  .  .  cover  all  possibilities, 

and  no  two  of  the  conclusions  X,  F,  Z, 

.  .  .  can  be  true  at  once. 

To  prove.    If  X,  then  A ; 
If  F,  then  B ; 
If  Z,  then  C,  etc. 
183 


184  APPENDIX 

Proof.     I.    If  JT,  then  not  F,  Z,  or  any  other  conclusion  (no 

two  can  be  true  at  once,  by  the  given). 
II.    .*.  not  B,   not   C,  not  any  condition   but   A 

(contraposite). 
III.    .*.  If  J$T,  then  A  (only  remaining  possibility). 

341.  Proofs  of  the  Obverse  and  the  Converse  of  a  Single 
Statement.  (1)  When  it  is  necessary  to  prove  the  obverse 
of  a  single  statement,  take  both  conditions,  and  show  that 
they  cannot  give  the  same  conclusion. 

If  A,  then  B.  Take  "  not  A"  and  show  that  it  cannot 
also  give  B. 

See  §  122 ;  in  this»case  A  represents  equal  angles,  B  rep- 
resents parallel  lines.  Then  "  not  A  "  represents  unequal 
angles,  and  the  reason  it  cannot  also  give  U,  that  is, 
parallel  lines,  is  that  there  can  be  but  one  parallel  through 
a  point. 

(2)  To  prove  the  converse  of  a  single  statement,  when 
that  is  necessary,  show  that  the  condition  and  the  conclu- 
sion give  one  and  the  same  thing. 

If  A,  then  B.     Show  that  B  gives  the  same  as  A. 

In  §  124,  if  it  is  to  be  proved  as  the  converse  of  §  121, 
the  proof  would  be  as  follows,  the  work  being  much 
abbreviated. 

If  A  (that  is,  the  angles  are  equal),  then  B  (that  is,  the 
lines  are  parallel). 

If  B  (that  is,  the  lines  are  parallel).  But  there  is  but 
one  parallel  through  the  point  in  question,  therefore  this  fig- 
ure is  identical  with  the  other,  and  so  the  angles  are  equal. 

Note  the  application  of  this  method  to  §  144. 

In  both  these  methods  the  proof  follows  because  of 
some  element  of  exclusion,  that  is,  some  statement  that 
there  can  be  but  one  of  a  certain  thing. 


APPENDIX  185 

342.  General  Condition.     In  many  of  the  theorems  of 
Geometry  the  hypothesis  contains  two  kinds  of  conditions, 
one  of  which  might  be  called  the  general  condition,  for  it 
states  the  kind  of  figure  about  which  the  statement  is  to 
be  made,  while  the  other  might  be  called  the  special  condi- 
tion, for  it  is  that  on  account  of  which  the  conclusion 
follows.     For  example,  in  Th.  I,  the  general  condition  is 
the  given  triangles ;  the  special  condition,  the  fact  that 
they  have  two  sides  and  the  included  angle  equal.     If 
the  converse,  obverse,  or  contraposite  is  to  be  stated,  the 
general  condition  should  be  left  just  as  in  the  original  state- 
ment, the  special  condition  and  the  conclusion  being  used. 

The  subject  could  be  made  very  complicated  by  going 
into  all  the  different  ways  of  using  the  two  conditions, 
but  the  statements  obtained  by  following  the  direction 
given  above  will  be  sufficient  for  all  practical  purposes. 

343.  Axioms.     The  axioms,  as  used  in  this  book,  are  not 
entirely  independent ;  for  example,  the  substitution  axiom 
could  be  derived  from  the  equality  axioms,  as  far  as  the 
equal  cases  are  concerned.     The  position  of  the  author  in 
thig  respect  has  been  defined  in  the  preface.     The  follow- 
ing axiom  can  be  derived  from  the  axiom  of  intersection, 
but  the  work  is  not  worth  while,  although  the  axiom  is 
needed  for  certain  work  in  the  theorems  and  exercises. 

Diagonal  Intersection  Axiom.  The  diagonals  of  any 
convex  quadrilateral  intersect  each  other  at  a  point 
within  the  quadrilateral. 

344.  Symmetry. 

(1)  With  respect  to  a  center.  Two  points  are  said  to  be 
symmetrical  with  regard  to  a  point  (or  center)  when  they 
are  on  opposite  sides  of  the  point  on  the  same  straight 
line,  and  at  the  same  distance  from  that  point.  Two 


186  APPENDIX 

figures  are  said  to  be  symmetrical  with  regard  to  a  center 
when  all  their  corresponding  points  are  respectively  sym- 
metrical with  regard  to  the  center. 

(2)  With  respect  to  an  axis.  Two  points  are  said  to  be 
symmetrical  with  regard  to  an  axis  when  they  are  on 
opposite  sides  of  a  line  (or  axis)  on  a  perpendicular  to 
the  axis,  and  equidistant  from  the  axis.  Two  figures  are 
said  to  be  symmetrical  with  regard  to  an  axis  when  any 
two  corresponding  points  are  symmetrical  with  regard  to 
that  axis. 

The  placing  of  figures  in  symmetrical  positions,  as  in 
Bk.  I,  Th.  VII  (§  105),  is  quite  common,  as  is  the  use  of 
symmetrical  points.  Note  also  the  figure  of  §  100,  and 
exercise  120. 

345.  Positive  and  Negative  Sects  and  Angles.     There 
are  many  theorems  in  Geometry  where  different  cases  of 
the  theorem  require  that  sects  or  angles  be  added  in  one 
case,  subtracted  in  a  second,  and  in  the  third  one  of  the 
things  added  or  subtracted  is  zero.     If  these  are  arranged 
in  one  general  statement,  it  can  be  shown  that  the  sects, 
or  angles,  are  all  positive  in  the  addition  case,  that  one 
pair  has  become  zero  in  the  zero  case,  and  then  negative  in 
the  subtraction  case.     Examples  of  this  can  be  found  in 
the  proof  of  Th.  VII,  Bk.  I  (§105),  in  §§  207,  214,  and 
215,  and  in  §§  240  and  244. 

346.  Distance.     Where  the  word  "  distance  "  is  used,  it 
always  means  the  shortest  possible  distance. 

The  distance  between  two  points  means  the  straight-line 
distance.  That  this  is  the  shortest  line  has  been  partly 
proved  in  §  109,  where  the  straight  line  between  two  points 
is  shown  to  be  less  than  any  broken  line  between  the 
points.  That  it  is  less  than  any  curved  line  between  the 


APPENDIX  187 

points  has  not  been  shown,  but  it  can  be  proved  quite 
easily  by  showing  that  the  shortest  line  between  two 
points  must  pass  through  any  point  on  the  straight  line 
between  those  points,  and  therefore  through  every  point 
on  the  straight  line. 

The  distance 'from  a  point  to  a  line  is  the  perpendicular 
from  the  point  to  the  line  (§  113). 

The  distance  from  a  point  to  a  circumference  is  the  sect 
from  the  point  to  the  circumference  on  the  line  through 
the  center  (§  174). 

347.  Limits.  There  is  some  doubt  as  to  the  perfect 
accuracy  of  any  of  the  proofs  for  the  first  limit  theorem ; 
the  following  proof  is  probably  as  free  from  objections 
as  any. 

I.  If  two  variables  approaching  limits  are  equal  for  all 
values,  their  limits  are  also  equal. 

Given.  v  =  vr;  v  =  L,  vf=L. 

To  prove.        L  =  L'. 

Proof.        I.    v  =  L  -  x. 

V'  =  L'—X',  where  x  and  xr  are   quantities, 

either    positive    or    negative,    which    can 

become  indefinitely  small,  but  cannot  equal 

zero. 

II.    v  =  vf  (given),  .  • .  L  —  x  =  L'  —  xf  (eq.  same), 

and  L  —  Lr  =  x  —  x'  (eq.  +  ,  —  eq.). 
III.  L,  Lf  and  L  —  Lf  are  constants;  x  and  x'  can 
each  be  made  less  than  any  constant  except 
zero.  If  L  —  Lr  is  not  zero,  x  and  x'  can 
each  become  less  than  l  (z,  —  i'),  and  then 
x  —  xr  would  be  less  than  L  —  Lf,  even 
though  the  signs  were  such  that  the  values 
of  x  and  x1  were  added. 


188  APPENDIX 

IV.    But  this  is  impossible,  since  L  —  L1  —  x  —  xf. 
.-.  L-L'=0. 

II.  (1)  If  a  variable  approaches  the  limit  zero,  its  quo- 
tient by  a  constant,  and  its  product  by  a  constant  (other  than 
zero),  approaches  zero. 

Given.  v  =  0  ;  c. 

To  prove.        -  =  0  ;  vc  =  0. 

Proof.  I.  Let  k  be  any  constant  quantity,  however 
small.  Then  ck  is  a  constant  quantity, 
and  v  can  become  less  than  ck. 

1) 

II.    But  if  v  <  ck,  then  -  <k  (uneq.  -5-  eq.)  . 

0 

HI.    And  -=£0,  since  v^O  (def.). 

0 

IV.    Since  -   is  less  than  k,  a  constant  quantity, 

0 

however  small,  but  is  not  zero,  .  •  .  -  =  0  (def.). 

c 

Similarly  v<?  =  0. 

(2)  If  a  variable  approaches  any  limit,  its  quotient  by  a 
constant,  and  its  product  by  a  constant  {other  than  zero), 
will  approach  the  quotient  of  its  limit  by  the  constant,  and 
the  product  of  its  limit  by  the  constant. 

Given.  v  =  L  ;  c. 

V       L 

To  prove.        c  =  c  ;  VC  =  LC. 
Proof.        I.    If  v  =  L,  then  L-v  =  0  (def.). 
II.    ,.()  = 


III.    .-.     =     (def.). 

Similarly  vc  =  Lc. 


APPENDIX  189 

III.  If  two  variables  are  proportional  to  two  constants, 
their  limits  are  proportional  to  the  same  constants. 

v      v' 

Given.  -  =  — ;  v  =  L,  v'  =  L  . 

c      c 

L       L' 

To  prove.        ~  =  ~J' 

c        c/ 

Proof.       I.    Since  v  =  L,  -  =  —  (II). 

c      c  ^ 

II.    Since  v'  =  L' ,  -7  =  ^- 

III.     But-  =  T-     •••-  =  ±7(0- 

c      c'  c       c' 

348.  Incommensurable  Case.  The  incommensurable 
case  of  Th.  V,  Bk.  II,  will  be  taken  as  an  example  of  the 
method  to  be  applied  to  all  theorems  having  the  two  cases. 

Let  the  central  angles  AOB  and  COD  in  circle  O  have 
no  common  divisor.  Suppose  an  exact  divisor  of  Z  AOB 
to  be  applied  to  Z  COD  as  of  ten*  as  possible,  leaving  a 
remainder  Z  XOD,  which  is  therefore  less  than  the  divisor 
used. 

Then  Z  AOB   and  Z  COX  have  a  common   divisor,  so 

Z  AOB     AB  ,      ^  ,1 

—  =  — -  by  the  commensurable  case. 
Z  cox     ex 

But,  if  the  divisor  of  Z  AOB  is  taken  smaller  and 
smaller,  that  is,  is  made  to  approach  the  limit  zero,  the 
remainder  Z  XOD,  being  still  smaller,  will  also  approach 
the  limit  zero.  Therefore  Z  COX  will  approach  the  limit 
Z  COD.  Also  its  arc,  CX,  will  approach  the  limit  CD. 

But,  since  the  variables  Z  COX  and  CX  are  proportional 
to  the  constants* Z  AOB  and  AB,  their  limits  are  also  pro- 
portional to  those  constants ;  that  is, =  ^=-  - 

Z  COD      CD 


190  APPENDIX 

This  method  of  proof  will  apply  to  Th.  II,  Bk.  Ill,  to 
Th.  I,  Bk.  IV,  and  in  fact  to  all  proofs  where  the  method 
of  a  common  divisor  is  used. 

349.  Similar   Figures.       Similar   figures   have   already 
been  defined  (  §  282j)  as  those  which  have  their  correspond- 
ing angles  equal,  and  their    corresponding  sides  propor- 
tional.    This,  of  course,  applies  only  to  polygons.     The 
following  definition  is  sometimes  given,  and  while  it  is 
not  as  convenient  for  use  in  Plane  Geometry,  it  has  the 
merit  of  applying  to  all  kinds  of  figures,  including  those 
of  Solid  Geometry. 

Two  figures  that  may  be  placed  in  a  pencil  (or  sheaf, 
in  Solid  Geometry)  of  lines,  so  that  all  pairs  of  correspond- 
ing points  of  the  figures  cut  the  respective  lines  of  that 
pencil  in  the  same  ratio,  are  similar. 

These  two  definitions  are  identical  in  result,  as  far  as 
polygons  are  concerned,  as  may  be  shown  by  proving  the 
two  following  statements : 

1.  If  two  polygons  that  are  mutually  equiangular  and 
have  their  corresponding  sides  proportional  are  placed  with 
one  pair  of  corresponding  sides  parallel,  the  polygons  lying 
on  the  same    side    of  those    lines,    the    lines  joining   their 
corresponding  vertices  will  form  a  pencil  which  is  cut  pro- 
portionally by  the  vertices  of  the  polygons. 

2.  If  two  polygons  lie  in  a  pencil  of  lines,  and  their 
vertices  cut  the  lines  proportionally ',  the  polygons  are  mutually 
equiangular,  and  their  corresponding  sides  are  proportional. 

350.  The  Evaluation  of  Pi.     The  ratio  of  the  circum- 
ference   of  a  circle   to  its  diameter   is  obtained  by  the 
use  of  the  ratios  of  the  perimeters  of  regular  inscribed 
and  circumscribed  polygons  of  a  large  number  of  sides  to 
the   diameter.      The   circumference   is   longer   than   the 


APPENDIX  191 

perimeter  of  any'  inscribed  polygon,  and  shorter  than 
the  perimeter  of  any  circumscribed  polygon,  so  that  if  the 
lengths  of  the  perimeters  of  two  regular  polygons  of  the 
same  number  of  sides,  one  inscribed,  the  other  circum- 
scribed, can  be  obtained  in  terms  of  the  diameter,  the 
length  of  the  circumference  will  lie  between  these  values, 
and  an  approximate  value  can  therefore  be  obtained. 

c1 
The  value   — ,  or  TT,  cannot  be  obtained  exactly,  for  it 

lias  been  proved  to  be  an  incommensurable  number.  It  is 
evident,  however,  that  the  value  obtained  will  be  more 
exact  as  the  polygons  used  have  a  larger  number  of  sides. 

The  approximation  is  started  by  finding  the  perimeters 
of  some  regular  polygon,  inscribed  and  circumscribed ; 
squares  or  hexagons  are  the  easiest  to  use.  The  perim- 
eters of  polygons  of  twice  the  number  of  sides  are  then 
worked  out,  and  the  doubling  is  continued  until  the  value 
is  found  to  the  required  degree  of  accuracy.  The  value 
has  been  carried  to  o'ver  700  decimal  places,  but  for  ordi- 
nary purposes  the  value  3. 14159+  is  sufficiently  accurate. 
3.1416  is  also  much  used,  and  3^  is  a  fair  approximation 
for  rough  work. 

The  following  theorems  give  the  material  for  the 
numerical  calculation : 

I.  The    perimeter    of   a    regular    inscribed    hexagon 
equals   3D;   the  perimeter   of  a  regular  circumscribed 
hexagon  equals   2 D v3. 

II.  If  the  perimeters  of  regular  inscribed  and  cir- 
cumscribed polygons  of  any  number  of  sides  are  known, 
the  perimeter  of  the  regular  circumscribed  polygon  of 
double  the  number  of  sides  equals  twice  their  product 
divided  by  their  sum. 

III.  If  the  perimeters  of  the  regular  inscribed  poly- 


192  APPENDIX 


ij>on  of  any  number  of  sides,  and  the  regular  circum- 
scribed polygon  of  double  that  number  of  sides,  are 
known,  the  perimeter  of  the  regular  inscribed  polygon 
having  double  the  first  number  of  sides  equals  the 
square  root  of  their  product. 

Applying  the  formula  in  II  to  the  regular  hexagons, 
calling  the  perimeter  of  a  regular  circumscribed  dodeca- 
gon P  : 

p=2(3D)(2W3)=  12W3  =12 


3D  +  2W3        3  +  2V3 
or,  p  =  (3.21539+)z>. 

Applying  the  formula  in  III  to  this  value  and  the  perim- 
eter of  the  regular  inscribed  hexagon,  and  calling  the 
perimeter  of  the  regular  inscribed  dodecagon  p  : 


Applying  these  methods  to  the  dodecagons,  the  perime- 
ters of  circumscribed  and  inscribed  regular  polygons  of  24 
sides  are,  respectively,  (3.15966-)Z)  and  (3.13263~)D. 

It  is  already  evident  that  the  circumference  of  the 
circle,  since  its  value  is  between  those  found,  must  be 
(3.1+)D,  and  a  continuance  of  the  work  will  determine 
additional  figures. 


YB   17314 


M306177 


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